\section{Introduction}
\label{sec:intro}

Semi-analytic models and numerical simulations predict that the first astrophysical objects formed at redshifts $z\gsim$ 20.   They form out of metal-free gas with inefficient fragmentation, thus numerical simulations predict that these first objects could be very massive ($M_\ast\gsim 100 M_\odot$), so-called 'Population III' (Pop III) stars \citep{ABN02, BCL02, YOH08}.  These stars would be short-lived, with lifetimes of $\sim 3$ Myr, but could produce ten times more ionizing photons per baryon than regular Population II stars \citep{Schaerer02, Schaerer03}.  Also, such star formation could have been somewhat synchronized, given that these early objects formed in highly biased and clustered environments.  Hence, Pop III stars could have a significant impact on subsequent generations of objects.

Indeed Pop III stars can strongly affect the progress of cosmological reionization \citep{HH03, Cen03a, Cen03b, WL03_postWMAP}, although their contribution is virtually unknown from first principles and is generally applied to simulations with a ``tuning knob" approach.  Thus ionizing photons from the first stars and lack thereof have been used to explain both early (e.g. \citealt{Cen03b}) and fairly late (e.g. \citealt{HB06}) reionization, depending on which scenario was favored when the works were published.  Therefore, it is quite important to gain physical insight into how the radiation of the first stars impacted their surroundings.

Radiative feedback can be either positive or negative, in that it can
enhance or suppress subsequent star--formation.  Positive feedback can
result when the enhanced free-electron fraction from ionizing photons (e.g. \citealt{OH02}) or hydrodynamical shocks \citep{SK87}
catalyzes the formation of molecular hydrogen (H$_2$). If the ionizing background ``turns-off" (resulting in so-called ``relic" HII regions), H$_2$ cooling can provide the dominant cooling channel at high-densities and low temperatures.
Conversely, negative feedback can result from heating by ionizing radiation which can photo-evaporate gas in low-mass halos \citep{Efstathiou92, BL99, Gnedin00filter, SIR04, Dijkstra04, MD08}.  Also, an active background of Lyman-Werner (LW) radiation (with photon energies in the 11.18--13.6eV range) can dissociate H$_2$, thus decreasing the gas's cooling capabilities (e.g., \citealt{HRL97, HAR00, CFA00, MBA01, WA07, Oshea08}).

Indeed, simulations find radiative feedback to be nuanced at very high redshifts.  Positive feedback dominates in flash ionized gas \citep{OShea05}, or gas exposed to a weak transient ultraviolet background (UVB) (\citealt{MBH06}, hereafter MBH06) such as might be present close to the edges of HII regions \citep{RGS02b, KM05}.  On the other hand, negative feedback in relic HII regions can occur in regions closer to the Pop III star (MBH06; \citealt{SU06, AS07, Yoshida07}).  Furthermore, radiative transfer effects can also impact the strength and sign of feedback (e.g. \citealt{ISR05, SU06, AWB07, Whalen08, WA08}).  Because of variations in halo mass, collapse redshift, central baryon density, stellar spectrum, and distance to the ionizing source, it is difficult to accurately deal with the large parameter space governing photoevaporation.

In MBH06, we attempted to tackle this issue with a statistical approach. By giving up on simulating ``realistic" (as much as this is possible given present uncertainties) relic HII regions left behind by deceased Pop III stars, we were able to explore larger swaths of parameter space and have a fairly large sample of second-generation objects.  We did this by applying various uniform ultraviolet (UV) and LW backgrounds on our entire simulation box, and statistically comparing the resulting evolution against the fiducial run without radiation.  Although this approach clearly does not realistically treat the photoionization around a particular source, it might provide a good statistical description of a ensemble of reionization studies.  Clearly, at the onset of early reionization not all halos lie within HII regions, as they do in our models. Our primary purpose is not to simulate early reionization or the photoevaporation of well-formed Pop III halos in detail (which would require radiative transfer) but to evaluate the impact of relic HII regions on the formation of halos at later times. Our uniformly illuminated boxes provide an ensemble of evaporated halos at high redshift that statistically sample the effects of early photoionization on the assembly of halos and formation of cold dense gas at lower redshifts.

However, since our analysis in MBH06 focused on a highly biased region which went non-linear at a high redshift, we could not follow the evolution to redshifts much lower than $z\sim20$. This meant that we could only extrapolate the observed trends suggesting that the feedback was transient.  \citet{OH03} claim that, even in the presence of a weak LWB, excess entropy will eventually suppress star formation in protogalaxies forming inside relic HII regions. In MBH06, we were unable to conclusively verify or refute this claim.

In this work, we extend the study by MBH06 by analyzing radiative feedback in a less-biased region, whose typical proto-galaxies form at lower redshifts, $z\sim$ 13--20.  Given that the 5-yr {\it WMAP} polarization data place a constraint on the reionization redshift (assuming instantaneous reionization) of $z=11.0 \pm 1.4$ \citep{Dunkley08}, such cosmological regions are likely to host the majority of feedback effects from Pop III stars, before a persistent UVB and/or a strong LWB become entrenched (e.g. \citealt{HRL96}).  We apply the same statistical approach as in MBH06, and study various combinations of transient UVBs and persistent LWBs.  As such, we attempt to provide a framework for analytically incorporating such radiative feedback into semi-analytic, and large-scale numerical and semi-numerical studies.

In \S \ref{sec:sims} we enumerate and describe the simulations which are used in this work. In \S \ref{sec:trans}, we study the initial radiative feedback caused by a transient UVB, adding a persistent LWB in \S \ref{sec:LW}.  Then in \S \ref{sec:pos}, we describe a related mechanism that results in eventual positive feedback, regardless of the strength of the UVB. In \S \ref{sec:ss}, we discuss the impact of neglecting self-shielding.  Finally, we offer our conclusions in \S \ref{sec:conc}.

Throughout this paper, we adopt the background cosmological parameters
($\Omega_\Lambda$, $\Omega_{\rm M}$, $\Omega_b$, n, $\sigma_8$, $H_0$)
= (0.7, 0.3, 0.047, 1, 0.92, 70 km s$^{-1}$ Mpc$^{-1}$), consistent
with the measurements of the power spectrum of CMB temperature
anisotropies by the first year of data from the {\it WMAP} satellite
\citep{Spergel03}.  Although the 5-yr data prefers a slightly lower value of $\sigma_8=0.82$ \citep{Dunkley08, Komatsu08} which somewhat delays structure formation, we keep the cosmology the same as in MBH06, to facilitate direct comparison.  Unless stated otherwise, we quote all quantities in comoving units.

\section{Simulations}
\label{sec:sims}

We use the Eulerian adaptive mesh refinement (AMR) code Enzo, which is
described in greater detail elsewhere \citep{Bryan99, NB99}.  Our
simulation volume is 1 $(h^{-1} ~ {\rm Mpc})^3$, initialized at $z_{\rm init}=99$ with
density perturbations drawn from the \citet{EH99} power spectrum.  Our root grid is
$128^3$. We have two additional static levels of refinement inside
a central 0.25 $h^{-1} ~ {\rm Mpc}$ cube.  In addition, grid
cells inside the central region are allowed to dynamically refine so
that the Jeans length is resolved by at least 4 grid zones and no grid
cell contains more than 4 times the initial gas mass element.  Each
additional grid level refines the mesh length of the parent grid cell
by a factor of 2.  We allow for a maximum of 10 levels of refinement
inside the refined central region, granting us a spatial resolution of
7.63 $h^{-1}~{\rm pc}$.  This comoving resolution translates to 0.36
$h^{-1}~{\rm proper~pc}$ at $z=20$.

As stated above, our simulation runs were set up to facilitate comparison with MBH06.  Users interested in details of the simulations are encouraged to consult MBH06 and \citet{MBA01}.
There is one notable difference between our runs here and those in MBH06: our central refined region for these runs is {\it not} centered on the highest density region of the box.  Instead is it chosen to be more typical of the regions expected to host the bulk of halos which form prior to any significant cosmological reionization.  Specifically, the density inside our central $(0.25 h^{-1} ~ {\rm Mpc})^3$ corresponds to a 0.75 $\sigma$ mass fluctuation of an equivalent spherical volume (in MBH06, we studied a 2.4 $\sigma$ region).  The less overdense region allows us to extend the analysis of MBH06 to lower redshifts, and test the robustness of the conclusions of MBH06 on a less-biased and more typical region.

As shown in Table \ref{tbl:runs}, we have performed four different
runs without a LW background, distinguished by the duration or
amplitude of the assumed UVB, and
eight additional runs that include an additional constant LW background.  Again, these runs are analogous runs to those in MBH06.
For the UV radiation we assume an isotropic background flux with a
$T=2\times10^4$K blackbody spectral shape, normalized at the hydrogen
ionization frequency, $h \nu_H$ = 13.6 eV.  This spectrum is softer than
the $\sim 8 \times 10^4$ K blackbody spectrum predicted to be produced
by Pop III stars (e.g. \citealt{Schaerer02}), which leads us to find a lower
reionization temperature (by about a factor of two) than we would have
found with a harder spectrum.  This results in outflow velocities (and therefore halo evacuation times)
which are also too low by this factor.

The values of $J_{21}^{UV}$
are shown in Table \ref{tbl:runs} in units of $10^{-21}{\rm
ergs~s^{-1}~cm^{-2}~Hz^{-1}~sr^{-1}}$.  The NoUVB\ run contains no
UV radiation, and serves mainly as a reference run. The Heat0.08\
and Heat0.8\ runs include a UVB with $J_{21}^{UV} = 0.08, 0.8$,
respectively.  The value of $J_{21}^{UV}=0.08$ was chosen to correspond
to the mean UV flux expected inside a typical HII region surrounding a
primordial star (e.g., \citealt{ABS05}).  As we do not include
dynamically expanding HII regions in our code, the Heat0.8\ and
Flash\ (in which the UVB is turned on for a negligibly short duration) runs can be viewed as extremes, corresponding to conditions
close to the center and close to the edge of the HII region,
respectively.
More generally, studying a range of values of $J_{\rm
UV}$ is useful, since the radiative properties of high-redshift sources are uncertain.
In the latter two runs, the UVB is turned on at
$z_{\rm UVB, on}=25$ and turned off at $z_{\rm UVB, off}=24.64$. This redshift range
corresponds to a typical theoretical stellar lifetime, $\sim3$ Myr, of
a $\sim 100 M_\odot$ primordial (Pop-III) star \citep{Schaerer02}.  The
flash ionization run, Flash, instantaneously sets the gas temperature
to T=15000 K and the hydrogen neutral fraction to $x_{\rm HI} = 10^{-3}$
throughout the simulation volume, but involves no heating thereafter.
Again, our adopted temperature is probably too low by a factor of two
compared to gas heated by Pop III stars.
  Finally, the eight runs in
the bottom half of the Table repeat pairs of the NoUVB\ and
Heat0.8\ runs with four different constant LW backgrounds
($J_{21}^{\rm LW}=$ 10$^{-4}$, 0.001, 0.01, and 0.1, normalized at 12.87eV in units of
$10^{-21}{\rm ergs~s^{-1}~cm^{-2}~Hz^{-1}~sr^{-1}}$, and assumed to be
frequency--independent within the narrow LW band).

\begin{table}
\vspace{0.2cm}
\caption{Summary of Simulation Runs}
\vspace{-0.2cm}
\label{tbl:runs}
\begin{center}
\begin{tabular}{ccccc}
\hline
Run Name & $J_{21}^{UV}$ &  $z_{\rm UVB, on}$ & $z_{\rm UVB, off}$ &   $J_{21}^{\rm LW}$  \\
\hline
\hline
\multicolumn{5}{c}{Runs without a LW background}\\
\hline
NoUVB      &   0   &    NA   &   NA    & 0 \\
Flash        &   --  &    25   &   25    & 0 \\
Heat0.08   &  0.08 &    25   &  24.62  & 0 \\
Heat0.8  &  0.8  &    25   &  24.62  & 0 \\
\hline
\multicolumn{5}{c}{Runs with a LW background}\\
\hline
LW10$^{-4}$\_NoUVB      &   0   &    NA   &   NA    & $10^{-4}$\\
LW10$^{-4}$\_Heat0.8  &  0.8  &    25   &  24.62  & $10^{-4}$\\
LW10$^{-3}$\_NoUVB      &   0   &    NA   &   NA    & $10^{-3}$\\
LW10$^{-3}$\_Heat0.8  &  0.8  &    25   &  24.62  & $10^{-3}$\\
LW10$^{-2}$\_NoUVB      &   0   &    NA   &   NA    & 0.01\\
LW10$^{-2}$\_Heat0.8  &  0.8  &    25   &  24.62  & 0.01\\
LW0.1\_NoUVB      &   0   &    NA   &   NA    & 0.1\\
LW0.1\_Heat0.8  &  0.8  &    25   &  24.62  & 0.1\\
\hline
\end{tabular}\\
\end{center}
\end{table}

We obtain the dark matter (DM) and baryonic mass of halos, in the same manner as MBH06.  Similarly, we use the fraction of
total gas within the virial radius which is cold and dense (CD),
$f_{\rm cd}$ as a proxy for the gas reservoir available to form stars.  By cold, we mean gas whose temperature is $<0.5 T_{\rm vir}$, where
$T_{\rm vir}$ is the halo's virial temperature (for how virial temperatures
are associated with halos in the simulation, see \citealt{MBA01}).  By
dense, we mean gas whose density is $> 10^{19}$ $M_\odot$ Mpc$^{-3}$
$\approx$ 330 cm$^{-3}$, roughly corresponding to the density at which
the baryons become important to the gravitational potential in the core,
taken to be an immediate precursor to primordial star formation
\citep{ABN02}.

Additionally, we discount halos which have been substantially
contaminated by the large (low-resolution) DM particles originally outside of our
refined region which enter it as it becomes nonlinear.  Specifically, we remove from our analysis halos with an average DM particle mass greater than $\sim$140\% of the refined region's
DM mass resolution, $747M_\odot$.  However, since there are small stocastic variations in the amount of contamination between the runs, we first apply this cut-off in the NoUVB\ run, and then remove the same halos from the analysis of the other runs.  This way, we study the same halos in each run.

Another possible source of
contamination arises from closely separated halos.  If some CD gas
belonging to a halo is within another halo's virial radius (most
likely in the process of merging), the other halo could undeservedly
be flagged as containing CD gas as well.  To counteract this, we set
$f_{\rm cd}=0$ for all halos without CD gas in their inner-most spherical shell.  CD gas invariably first forms close to the halo's core, where the cooling times are the shortest; thus halos which contain CD gas at large radii but none in their core are most likely in the process of merging with a halo with genuine CD gas. Note that this algorithm is different than the one used in MBH06, where we discriminated based on mass and separation between halos.  In this work such contamination was more prevalent and so we devised what proved to be a more robust means of discrimination.

Finally, we note that we do not include HD cooling in these simulations.  As is shown in \citet{McGreer08} and \citet{Yoshida07}, HD cooling is important in primordial halos only in the density range $10^2 - 10^6$ cm$^{-3}$, where H$_2$ can bring the temperature below about 200 K.  Therefore it does not change the amount of CD gas, our primary diagnostic (although it would lead to relatively small changes in the predicted masses of the resulting stars).

\section{Transient Feedback Following a Transient UVB}
\label{sec:trans}

\begin{figure*}
\vspace{+0\baselineskip}
{
\includegraphics[width=0.245\textwidth]{early2_profile_z24.62.eps}
\includegraphics[width=0.245\textwidth]{early2_profile_z23.eps}
\includegraphics[width=0.245\textwidth]{early2_profile_z21.eps}
\includegraphics[width=0.245\textwidth]{early2_profile_z19.eps}
\vskip0.0pt
}
{
\includegraphics[width=0.245\textwidth]{early2_profile_z24.62.Species.eps}
\includegraphics[width=0.245\textwidth]{early2_profile_z23.Species.eps}
\includegraphics[width=0.245\textwidth]{early2_profile_z21.Species.eps}
\includegraphics[width=0.245\textwidth]{early2_profile_z19.Species.eps}
}
 \caption{
Spherically averaged radial
profiles of the same halo in the NoUVB\ ({\it solid lines}), Heat0.08\ ({\it dashed lines}) and
Heat0.8\ ({\it dotted lines}) simulation runs.  This halo was first able to form CD gas at $z=$ 20, 21, 19 in the NoUVB, Heat0.08, and Heat0.8\ runs, respectively.
The top and bottom pairs of figures show snapshots at $z =$ 24.62, 23, 21, 19 ({\it left to right}).
The halo mass in the NoUVB\ run increases from M=$4.2\times10^5 M_\odot$ at $z=24.62$, to M=$1.4\times10^6 M_\odot$ at $z=19$.
All quantities are shown in proper (not comoving) units.
The {\it Upper panels} show the hydrogen density, mass-weighted gas
temperature, gas cooling time, and radial velocity ({\it clockwise
from upper left}).  The {\it Bottom panels} show mass fractions of HI,
HII, H$_2$, and the number fraction of $e^-$ ({\it clockwise from
upper left}).
\label{fig:early_profiles}
}
\vspace{-1\baselineskip}
\end{figure*}

We first wish to confirm whether our conclusions in MBH06 also apply to this new sample of halos, which form at later times in a less overdense region of the universe.  In MBH06, we found the following.  The UVB ionizes and heats the gas in our simulation box.\footnote{It is important to note that we are operating in the optically thin limit; the accuracy and impact of that approximation is discussed in section~\ref{sec:ss}.}  Gas inside halos which used to be at or close to hydrostatic equilibrium feels an increased pressure gradient from the rise in temperature, resulting in an outward--moving pressure shock.  Once the UVB is turned off, the gas quickly cools to $T\sim10^3$ K through a combination of atomic, molecular hydrogen and Compton cooling. The molecular hydrogen formation time is shorter than the recombination time and a large amount of molecular hydrogen is produced, $x_{\rm H_2} \sim$ few $\times 10^{-3}$, irrespective of the density and temperature (see also \citealt{OH02}).\footnote{\citealt{Glover08} have recently shown that including cooling due to excitations of $H_2$ through collisions with protons and electrons can lead to a lower molecular hydrogen fraction by a factor of 2.  This would result in a longer delay in star-formation than shown here.}  The pressure shock starts to dissipate and gas expelled from the center of the halo starts falling back in (for a more complete description, readers are encouraged to see MBH06).

Quantitatively, we concluded that the net impact of a transient UVB on star-formation could be understood simply in terms of the molecular hydrogen cooling time-scale:

\begin{align}
\label{eq:t_H2}
t_{\rm H_2} &= \frac{1.5 k_B T}{\Lambda_{\rm H_2}} \frac{n_g}{n_{\rm HI} n_{\rm H_2}}\\
&\approx  \frac{1.5 k_B T}{\Lambda_{\rm H_2}} \frac{1}{n_g x_{\rm H_2}}\\
&\approx 4 \left( \frac{T}{10^3 K} \right)^{-2.5}
                   \left( \frac{x_{\rm H_2}}{3 \times 10^{-3}} \right)^{-1}
                   \left( \frac{n_g}{1 \rm cm^{-3}} \right)^{-1} {\rm Myr}~ , \nonumber
\end{align}

\noindent Here, $k_B$ is the Boltzmann constant, $T$ is the temperature,
$\Lambda_{\rm H_2}$ is the H$_2$ cooling function, $x_e$ is the free
electron number fraction, $n_g$, $n_{\rm HI}$, and $n_{\rm H_2}$ are
the number densities of all baryons and electrons, neutral hydrogen,
and H$_2$, respectively, and the second equality is accurate shortly after $z_{\rm UVB, off}$, since the recombination times at high redshifts and inside halos are very short.

We found that the net {\it delay} in star-formation caused by the transient UVB is simply the ratio of the H$_2$ cooling time in the run with the UVB, to the H$_2$ cooling time in the run without UV heating.  This ratio is calculated near the halo core, shortly after the UVB disappears (when the shock first starts to dissipate and the gas behind it starts infalling again).  For example, the delay in star-formation of a halo in the Heat0.8\ run with respect to the same halo in the NoUVB\ run is:

\begin{equation}
\label{eq:delay}
f_{\rm delay} \sim
\frac{t_{\rm H_2}^{\rmHeat0.8}}{t_{\rm H_2}^{\rmNoUVB}} \sim
\left( \frac{T^{\rmNoUVB}}{T^{\rmHeat0.8}} \right)^{2.5}
\left( \frac{n^{\rmNoUVB}_g}{n^{\rmHeat0.8}_g} \right)
\left( \frac{x^{\rmHeat0.8}_{\rm H_2}}{x^{\rmNoUVB}_{\rm
H_2}} \right)^{-1} ~.
\end{equation}

We will now demonstrate that eq. (\ref{eq:delay}) is equally adept at quantifying feedback in our new simulations.  First, we show the spherically averaged radial profiles of the same (typical) halo in the NoUVB\ ({\it solid lines}), Heat0.08\ ({\it dashed lines}) and Heat0.8\ ({\it dotted lines}) simulation runs in our Figure \ref{fig:early_profiles}.  This halo, which has a mass of $4\times10^5 M_\odot$ at $z=24.62$, was first able to form CD gas at $z=$ 20, 21, 19 in the NoUVB, Heat0.08, and Heat0.8\ runs, respectively. The top and bottom pairs of figures are snapshots at $z =$ 24.62, 23, 21, and 19 ({\it left to right}). The {\it upper panels} show the hydrogen density, mass-weighted gas temperature, gas cooling time, and radial velocity ({\it clockwise from upper left}).  The {\it bottom panels} show mass fractions of HI, HII, H$_2$, and the number fraction of $e^-$ ({\it clockwise from upper left}).

The profiles confirm the qualitative story line from the beginning of this section. The outward-moving pressure shock is evident in both UVB runs from the density and velocity panels at $z=z_{\rm UVB, off}$, just before the UVB was turned off.  This gas has already cooled to $T=10^3$ K and started to collapse back onto the halo only a few Myr afterward at $z=23$.  Note that the outflow and the resulting suppression of gas near the core is stronger in the run with a stronger UVB.  Halos in the runs with a UVB also experience a strong boost in the H$_2$ fraction following $z_{\rm UVB, off}$, with the increase being insensitive to the strength of the UVB.  Formation of CD gas in this halo is notably delayed until $z=19$ in the Heat0.8\ run, while being mildly expedited in the Heat0.08\ run.  Can this behavior be understood through eq. (\ref{eq:delay})?

Let us focus on the output we have shortly after $z_{\rm UVB, off}$, at $z=23$.  We can note that at this redshift, the pressure-driven shock has almost dissipated and reached zero velocity with gas recommencing infall behind the shock.  Plugging in appropriate values into eq. \ref{eq:delay} from the inner annulus of the halo, we obtain values of $f_{\rm delay}\approx$ 0.5 and 1.5 for the Heat0.08\ and Heat0.8\ runs, in good agreement with the simulation outcomes.  This behavior again supports our conclusions in MBH06, where the same analysis held for a different, randomly-chosen halo, and implies that we understand the physical picture associated with feedback in relic HII regions.

In order to quantify the suppression of CD gas by a transient UVB on our entire sample of halos,
 we define (as in MBH06) the cumulative, fractional suppression of the number of newly formed halos as
\begin{equation}
\label{eq:delN}
\delta_{N, {\rm cd}}(z) \equiv \frac{N_{\rm cd}^{{\rm run}i}(z) - N_{\rm cd}^{{\rm run}i}(z_{\rm UVB, on})}{N_{\rm cd}^{\rm
NoUVB}(z) - N_{\rm cd}^{\rm NoUVB}(z_{\rm UVB, on})} - 1 ~ ,
\end{equation}
where $N_{\rm cd}^{\rm NoUVB}(z)$ and $N_{\rm cd}^{{\rm run}i}(z)$ are the total
number of halos with CD gas at redshift $z$ in the NoUVB\ run and
some given run $i$, respectively.  This expression is well--defined
for $N_{\rm cd}^{\rm NoUVB}(z)$ $>$ $N_{\rm cd}^{\rm
NoUVB}(z_{\rm UVB, on})$; for $N_{\rm cd}^{\rm NoUVB}(z)$ = $N_{\rm
cd}^{\rm NoUVB}(z_{\rm UVB, on})$, we set $\delta_{N, {\rm cd}}(z) \equiv 0$.  Note that by
definition, $N_{\rm cd}^{\rm NoUVB}(z_{\rm UVB, on})$ = $N_{\rm cd}^{{\rm run}i}(z_{\rm UVB, on})$.

Similarly, we define the cumulative, fractional suppression of the CD
gas mass as
\begin{equation}
\label{eq:delM}
\delta_{M, {\rm cd}}(z) \equiv \frac{M_{\rm cd}^{{\rm run}i}(z) - M_{\rm cd}^{{\rm run}i}(z_{\rm UVB, on})}{M_{\rm cd}^{\rm
NoUVB}(z) - M_{\rm cd}^{\rm NoUVB}(z_{\rm UVB, on})} - 1 ~ ,
\end{equation}
where $M_{\rm cd}^{\rm NoUVB}(z)$ and $M_{\rm cd}^{{\rm run}i}(z)$ are the total mass
of CD gas at redshift $z$ in the NoUVB\ run and some given run $i$,
respectively.  The total CD gas mass is obtained by merely summing the
CD gas masses for all of the halos in the simulation.  As for
equation~(\ref{eq:delN}), this expression is well--defined for $M_{\rm
cd}^{\rm NoUVB}(z)$ $>$ $M_{\rm cd}^{\rm NoUVB}(z_{\rm UVB, on})$, and
for $M_{\rm cd}^{\rm NoUVB}(z)$ = $M_{\rm cd}^{\rm
NoUVB}(z_{\rm UVB, on})$, we set $\delta_{M, {\rm cd}}(z) \equiv 0$.

\begin{figure}
\myputfigure{raw.eps}{3.3}{0.5}{.}{0.}  \caption{Values of
$M_{\rm cd}^{{\rm run}i}(z)$ ({\it top panel}) and $N_{\rm cd}^{{\rm run}i}(z)$ ({\it bottom panel}) as
defined in equations (\ref{eq:delM}) and (\ref{eq:delN}).  The results
are displayed for the NoUVB\ ({\it crosses}), Flash\ ({\it
dashes}), Heat0.08\ ({\it triangles}), and Heat0.8\ ({\it
squares}) simulation runs.
\label{fig:NM}}
\vspace{-1\baselineskip}
\end{figure}

Equations (\ref{eq:delN}) and (\ref{eq:delM}) provide an estimate of
how the CD gas has been affected by the presence of a UVB, {\it
following} the turn-on redshift of the UVB, $z_{\rm UVB, on}$ (the values at
$z_{\rm UVB, on}$ are subtracted in order to provide a more sensitive measure
of {\it relative} changes of CD gas).  As defined above, $\delta_{N, {\rm cd}}(z) = 0$
and $\delta_{M, {\rm cd}}(z) = 0$ if the UVB has no effect.  If the effect of a UVB is
positive, resulting in positive feedback, $\delta_{N, {\rm cd}}(z)$ and $\delta_{M, {\rm cd}}(z)$ would be
positive.  If the effect of the UVB is negative, $\delta_{N, {\rm cd}}(z)$ and $\delta_{M, {\rm cd}}(z)$
would be negative.

In Figure \ref{fig:NM}, we plot the values of $M_{\rm cd}^{{\rm run}i}(z)$ ({\it top
panel}) and $N_{\rm cd}^{{\rm run}i}(z)$ ({\it bottom panel}) in our four
simulation runs with no LW background: ${\rm run}i$ = NoUVB\ ({\it crosses}), Flash\
({\it dashes}), Heat0.08\ ({\it triangles}), and Heat0.8\
({\it squares}).  We note that the number of halos with CD gas increases in all runs until $z\sim15$, when our central refined regions starts becoming more non-linear.  At these lower redshifts, halos become contaminated with large DM particles entering from outside the refined region and we remove them from our analysis sample (see \S \ref{sec:sims}).

The corresponding values of $\delta_{M, {\rm cd}}(z)$ and $\delta_{N, {\rm cd}}(z)$ are
plotted in Figure \ref{fig:delta} in the top and bottom panels,
respectively. Although some of the notable fractional
changes shown in Figure \ref{fig:delta} might appear statistically
insignificant due to the small number statistics inferred from Figure
\ref{fig:NM}, it should be noted that these runs are not uncorrelated
experiments.  In other words, each of our runs in Table \ref{tbl:runs}
is seeded with the same initial conditions, and so small relative
changes compared to the NoUVB\ run are significant (i.e. the errors
are not Poisson).  Nevertheless, there are some minor differences between the runs in the amount of ``contamination'' by DM particles outside of our refined region.

\begin{figure}
\myputfigure{delta_cum_sup.eps}{3.3}{0.5}{.}{0.}  \caption{
Values of $\delta_{M, {\rm cd}}(z)$ ({\it top panel}) and $\delta_{N, {\rm cd}}(z)$ ({\it bottom panel}) as defined
in equations (\ref{eq:delM}) and (\ref{eq:delN}).  The results are
derived from Figure~\ref{fig:NM} and displayed for the Flash\ ({\it
dashes}), Heat0.08\ ({\it triangles}), and Heat0.8\ ({\it
squares}) simulation runs.
\label{fig:delta}}
\vspace{-1\baselineskip}
\end{figure}

In MBH06, we presented two main inferences regarding the effects of a transient UVB (aside from the delay due to cooling times discussed above): (1) a ``critical'' background value of $J_{21}^{UV} \sim 0.1$ separated positive from negative feedback (stronger backgrounds result in negative feedback, while weaker backgrounds result in positive feedback; see the discussion surrounding eq. \ref{eq:delay} above); (2) the feedback effects, regardless of the sign, were transient and started to ``disappear'' by $z\sim18$. However, we were unable to evolve our simulations below $z<18$, since the central refined region became too non-linear and further progress was computationally prohibitive.  Thus point (2) was an inference based on a figure analogous to Fig. \ref{fig:delta}.

In this work we present results from a less biased region and are able to go to lower redshifts, $z\sim 13.5$.  Thus we can test both of the conclusions above and see if they are sensitive to the large-scale halo environment and redshift.  From figure \ref{fig:delta}, we see that {\it both} conclusions are valid for these simulations as well.  In other words, our transient (corresponding to the expected lifetime of a Pop III star) UVB results in transient feedback which disappears after $\sim30$\% of the Hubble time, with $J_{21}^{UV}\sim0.1$ separating net positive and negative feedback. {\it Thus, these conclusions seem to be fairly insensitive to the large-scale overdensity of the simulated region, and by extension to the redshift-dependent properties of halos at $z_{\rm UVB, on}$.}

It is important to note a new result evident in Fig. \ref{fig:delta}: {\it all runs with a transient UVB experience eventual positive feedback at $z\lsim15$}. This is true regardless of the strength of the UVB.  We will examine this interesting result further in \S \ref{sec:pos}.

\subsection{Adding a Persistent LW Background}
\label{sec:LW}

Next, we investigate how the combination of transient UV heating and a persistent LW background impact star formation.  This is a more physically-relevant scenario, since a LW background is likely established before a large fraction of the Universe has been reionized \citep{HAR00}.

Photons in the LW band dissociate H$_2$ molecules; thus by definition they provide additional negative feedback, undercutting some of the enhancement in the H$_2$ abundances in relic HII regions.  Negative feedback from a LW background kicks in when the H$_2$ dissociation timescale becomes shorter than the H$_2$ formation timescale.  Since the formation timescale is inversely proportional to gas density, whereas the dissociation timescale is independent of density, the density decrease caused by UV heating should make halos more susceptible to the negative feedback of a LW background \citep{OH03} \footnote{Note that the temperature increase associated with a UVB does not contribute much this combined negative feedback because the ionized gas is able to Compton cool down to the virial temperatures of these minihalos.}.  We investigate these processes further below.

\begin{figure}
\vspace{+0\baselineskip}
\myputfigure{LW_delta_cum_sup.eps}{3.3}{0.5}{.}{0.}
\caption{
Suppression of cold dense gas in halos in simulations runs that include a
persistent LW background. The LW background had specific intensities
of $J_{\rm LW}=10^{-4}, 10^{-3}, 0.01$, or $0.1$ (normalized at 12.87 eV, in
units of $10^ {-21}{\rm ergs~s^{-1}~cm^{-2}~Hz^{-1}~sr^{-1}}$). Each
of these three LW backgrounds is applied to both our NoUVB\ and
Heat0.8\ runs at $z=24.62$, and is subsequently left on.  Values
of $\delta_{M, {\rm cd}}(z)$ ({\it top panel}) and $\delta_{N, {\rm cd}}(z)$ ({\it bottom panel}) are
shown, as defined in equations (\ref{eq:delM}) and (\ref{eq:delN}).
\label{fig:LW_delta}}
\vspace{-1\baselineskip}
\end{figure}

Specifically, in Fig. \ref{fig:LW_delta}, we plot values of $\delta_{M, {\rm cd}}(z)$ ({\it top panel}) and $\delta_{N, {\rm cd}}(z)$ ({\it bottom panel}) for our runs including a LW background.  The LW background turns on at $z=24.62$ and remains on.  At the simplest level, we can see the suppression of molecular hydrogen (and hence cooling) due to the LW background by the more negative values of both $\delta_{M, {\rm cd}}(z)$ and $\delta_{N, {\rm cd}}(z)$ at fixed redshift.

Does this panel show evidence for the {\it additional} negative impact of the transient UVB discussed above?  Without UV heating, a LW background with a specific intensity of $J_{21}^{\rm LW}=0.01$ is needed for notable negative feedback (see the blue empty triangles at $z\leq14$ in the lower panel).  This value is similar to the one found in MBH06, where we showed that by equating ${\rm H_2}$--cooling and ${\rm H_2}$--photodissociation one expects this critical LW background to scale as $J_{21}^{\rm LW} \propto n_g f_{\rm H2}/T$.\footnote{Since the mean density of gas collapsing into halos in these simulations is less than a factor of two smaller than those in MBH06, we roughly expect this critical value of $J_{21}^{\rm LW}$ to also be less than a factor of two smaller (at least in the linear regime).}
  When the UV heating is added, this critical value of $J_{21}^{\rm LW}$ causing negative feedback decreases by a factor of $\sim10$ to $J_{21}^{\rm LW}\sim10^{-3}$ (see the green solid circles in lower panel at $z\leq14$).  This decrease seems to confirm the above arguments.  However, such an interpretation is too simplistic and the negative impact of the LWB can be ameliorated by positive feedback inside the HII region at lower redshifts, as we shall see in \S \ref{sec:pos}.

From Fig. \ref{fig:LW_delta} one can also note that a value of $J_{21}^{\rm LW}\sim 10^{-3}$--$10^{-2}$ separates feedback regimes dominated by a LW background from those dominated by our transient UVB.   This can be seen by the fact that the amount of suppression differs between the NoUVB and Heat0.8 cases at low values of the LW background (i.e. for $J_{21}^{\rm LW} < 10^{-3}$), while at large values of $J_{21}^{\rm LW}$, the amount of suppression is independent of the UVB.  Near the threshold value of $J_{21}^{\rm LW}$, negative feedback transitions from being UV heating dominated (at $z \gsim 17$, or $\lsim 100$ Myr after $z_{\rm UVB, off}$) to being LW background dominated ($ z \lsim 17$, or $\gsim 100$ Myr after $z_{\rm UVB, off}$).  As highlighted in MBH06, this ``transition'' behavior can be understood as the
combined result of two effects: the UV heating is turned off, and its
impact is transient, while the critical LW
background scales roughly with the inverse of the density
\citep{HAR00, OH03} and hence a fixed LW background will have a larger
impact at lower densities or decreasing redshifts.

\section{Eventual Positive Feedback}
\label{sec:pos}

So far, we have not focused on the interesting new result hinted by the previous figures (e.g. see the $z\lsim14.5$ points in Fig. \ref{fig:delta}).  This figure shows evidence of delayed (approaching our lowest redshifts) {\it positive} feedback in {\it all} runs with a UVB.  This result is somewhat surprising given that the Heat0.8 run shows strong negative feedback initially after $z_{\rm UVB, off}$. In fact, as was already pointed out when discussing Fig. \ref{fig:delta}, the initial feedback, whether positive or negative, fades away in {\it all} runs by $z\sim17$.

In Fig. \ref{fig:fcoll}, we show the total gas fractions ({\it top panels}) and CD
gas fractions ({\it bottom panels}) from halos at $z=13.5$.  The positive feedback mentioned above is immediately evident from this figure, and shows explicitly the masses of the affected halos.  In particular, the NoUVB\ run at $z=13.5$ has 3 {\it fewer} halos hosting CD gas than the Flash\ and Heat0.08\ runs, and 2 {\it fewer} than even the Heat0.8\ run (see also Fig. \ref{fig:NM}).  These are some of the smallest and youngest of such halos at these redshifts.  We also verify that these halos formed in less biased and less overdense regions than the halos which didn't experience such positive feedback.

\begin{figure*}
\vspace{+0\baselineskip}
{
\includegraphics[width=0.45\textwidth]{nouvb_gas_fracts_z13.50.eps}
\includegraphics[width=0.45\textwidth]{flash_gas_fracts_z13.50.eps}
\includegraphics[width=0.45\textwidth]{heat0.08_gas_fracts_z13.50.eps}
\includegraphics[width=0.45\textwidth]{heat0.8_gas_fracts_z13.50.eps}
}
\caption{
\label{fig:fcoll}
Total gas fractions ({\it top panels}) and cold, dense
gas fractions ({\it bottom panels}) as a function of total halo mass
at redshift $z=13.5$, the lowest redshift output of our simulations.
The four panels correspond to the NoUVB\ ({\it top left}), Flash\
({\it top right}), Heat0.08\ ({\it bottom left}), and Heat0.8\
({\it bottom right}) simulation runs. Note that all runs with a UVB have more halos with CD gas than the NoUVB\ run.}
\vspace{+1.5\baselineskip}
\end{figure*}

What is the physical cause of such delayed positive feedback?  Before we attempt to answer this question, let us briefly review the sequence of feedback-related events arising from a transient UVB:
\begin{enumerate}
\item When the UVB turns on, gas gets ionized and heated to temperatures $\gsim10^4$ K.
\item The temperature increase sets-off an outward moving pressure shock in the cores of halos, where density profiles have already steepened as gas approached hydrostatic equilibrium.
  This pressure shock smooths out the gas distribution and leads to a decrease in gas density in the cores of halos.
\item Once the UVB is turned off, the gas rapidly cools to $T\sim10^3$ K through a combination of atomic, molecular hydrogen and Compton cooling.  This temperature approximately corresponds to the gas temperature at the virial radius of such a proto-galactic, molecularly-cooled halo, thus effectively neutralizing the impact of temperature change on feedback.
\item A large amount of molecular hydrogen is produced, $x_{\rm H_2} \sim$ few $\times 10^{-3}$, irrespective of the gas density and temperature.
\item The pressure-shock begins to dissipate and gas with a newly enhanced H$_2$ abundance starts falling back onto the partially evacuated halo.
\end{enumerate}

It is the balance between the density depletion in step (ii) (which is dependent on the UVB intensity) and the enhanced cooling capability of the gas from step (iv) (which is independent of the UVB intensity), which determines the transient feedback outcomes discussed in \S~\ref{sec:trans}.  But what happens to gas which is not in the cores of already-formed halos with a steepened density profile?  In the extreme case of homogeneous gas, the UVB-induced temperature increase in (i) {\it does not translate into a pressure shock}.  Thus, there is no density depletion from step (ii).  In the more realistic case of a clumpy gas distribution, pressure gradients will still form and smooth out the gas distribution, but the resulting depletion will be less than in the cores of halos in hydrostatic equilibrium.  However, such lower-density gas outside of halos will still be able to cool to temperatures of $\sim10^3$ K since the Compton cooling time is independent of density, and at these high redshifts is over a factor of two shorter than the recombination time at mean density [in fact, these lower density regions take longer to recombine and hence can Compton-cool for a longer period reaching lower temperatures \citep{OH02}; see also equations (2) and (3) in MBH06].  Likewise, step (iv) mentioned above also applies to such lower density gas, with the transient background resulting in an enhanced molecular hydrogen abundance. Thus, we expect gas which has not yet formed a self-similar (e.g. \citealt{NFW96}) halo profile to experience weaker negative feedback [i.e. step (ii)], and the same amount of positive feedback [i.e. step (iv)], when exposed to identical transient UVBs.

This conclusion is perhaps a bit counterintuitive, since many cosmological radiative feedback studies find that feedback is a strong function of halo mass, with more massive, earlier forming halos being {\it less} susceptible to negative feedback \citep{Efstathiou92, BL99, Gnedin00filter, SIR04, KI00, Dijkstra04, MD08}.  However, there are two important distinctions here: we are discussing feedback resulting from a {\it transient} UVB; and we are comparing halos which have not yet formed and set up a self-similar gas profile at $z_{\rm UVB, off}$ to those which have already formed, rather than comparing {\it within} each of these two groups.

We also see some evidence of these trends in Fig. \ref{fig:ratios}, where we plot the ratio (in runs with a UVB over the fiducial NoUVB\ run) of the ${\rm H_2}$ fractions ({\it red squares}), the temperatures ({\it blue triangles}), and the average gas densities ({\it black circles}) within the central 15pc (proper) of halos.
These quantities are shown shortly after $z_{\rm UVB, off}$, at $z=23$.  As already noted in \S \ref{sec:trans} and in (iv) above, the ${\rm H_2}$ fraction tends to increase by the same factor, regardless of the nature of the heating. As a result, the sign of the overall feedback (negative or positive) is determined primarily by whether or not the depression of the gas density overcomes the positive effects of this H$_2$ enhancement. The density depletion does depend strongly on the type and amount of heating.
  This depletion, shown in black circles, is of course stronger for the Heat0.8\ than the Heat0.08\ run.  Thus halos in the former initially experience strong negative feedback and in the latter negligible or weak positive feedback, as we've seen in \S \ref{sec:trans} and as noted by the green crosses in the figure.  Note however that the difference in the amount of gas depletion between these two heating runs decreases with decreasing halo mass as one begins being affected by Jeans/filtering scale effects\footnote{Note that accurately modeling Jeans smoothing requires very high resolution in both dark matter and gas \citep{NBM09}. These small scales are shown in Fig. \ref{fig:ratios} merely to show qualitative trends.}. Following the above argument, this trend should continue and density depletion should decrease in regions which are less evolved (i.e. the black points should approach unity as the halo mass is decreased below the values shown in the figure).  It is these regions which become the progenitors of our halos experiencing eventual positive feedback.
  Unfortunately, at $z=23$, these regions are too unevolved to be identified by our halo finding algorithm and thus do not appear in Fig. \ref{fig:ratios}.

\begin{figure*}
\vspace{+0\baselineskip}
{
\includegraphics[width=0.45\textwidth]{ratios_heat0.08_z23.eps}
\includegraphics[width=0.45\textwidth]{ratios_heat0.8_z23.eps}
}
\caption{
\label{fig:ratios}
Ratio of the ${\rm H_2}$ fractions ({\it red squares}), the temperatures ({\it blue
triangles}), and the average gas densities ({\it black circles}) within the central 15pc (proper) of halos.  Ratios of these quantities in the Heat0.08\ and Heat0.8\ runs over the fiducial NoUVB\ run are shown in the left and right panels, respectively, at $z=23$.  The delay factors from eq. (\ref{eq:delay}) are also plotted as green crosses.}
\vspace{+1.5\baselineskip}
\end{figure*}

This eventual positive feedback should be quite sensitive to the LWB.  Indeed, looking at the bottom panel of Fig. \ref{fig:LW_delta}, we see that this feedback effect is wiped-out if $J_{21}^{\rm LW}\gsim10^{-3}$.  It is likely that such modest values of $J_{21}^{\rm LW}$ are reached well before the majority of the Universe is ionized \citep{HAR00}, though the buildup of the LWB can be slowed by an increasing optical depth to LW photons from the enhanced H$_2$ abundances in relic HII regions \citep{JGB07}, or in shells surrounding HII regions (\citealt{RGS01, KM05}; though see \citealt{KYSU04}).  Thus studying the true physical importance of this positive feedback mechanism would have to be done with self-consistent cosmological simulations of reionization.

Now, let us look at specific examples by studying the evolution of one such halo experiencing eventual positive feedback.  In Figure \ref{fig:NoUVB_vs_Heat}, we plot the same quantities as in Fig. \ref{fig:early_profiles}, but for a halo which experiences positive feedback in the Heat0.8\ run.  Note that, unlike the more evolved halos which are subjected to the transient feedback and have already formed at $z_{\rm UVB, off}$, the earliest snapshot we have of this halo is at $z=21$. Solid and dashed curves correspond to the NoUVB\ and Heat0.8\ runs, respectively.

It is important to note that this halo forms considerably after $z_{\rm UVB, off}$, and thus the ``density depletion" seen in the figure at $z\gsim 17$ is of a different nature than discussed in \S \ref{sec:trans} and Fig. \ref{fig:early_profiles}.  Namely, this gas wasn't evacuated from the halo center with a pressure shock.  Instead, these late forming halos start pulling in baryons when their virial temperature exceeds that of the gas temperature.  The heating and subsequent Compton cooling following the transient UVB sets the IGM gas temperature in these runs at $T\sim10^3$ K.  Thus baryonic infall onto newly formed halos can commence when $T_{\rm vir} \gsim 10^3$ K.  Since the IGM is cooler in the NoUVB\ run, gas there gets a ``head-start'' in accreting onto DM halos.  However, as is evident in the figure, the enhanced H$_2$ cooling channel in the Heat0.8\ run allows the gas to quickly catch-up and surpass its twin in the NoUVB\ run.  Thus by redshift $z=14$, the halo has formed CD gas in the Heat0.8\ run, but not in the NoUVB\ run.

\begin{figure*}
\vspace{+0\baselineskip}
{
\includegraphics[width=0.245\textwidth]{NoUVB_vs_Heat_z21.eps}
\includegraphics[width=0.245\textwidth]{NoUVB_vs_Heat_z19.eps}
\includegraphics[width=0.245\textwidth]{NoUVB_vs_Heat_z17.eps}
\includegraphics[width=0.245\textwidth]{NoUVB_vs_Heat_z14.eps}
\vskip0.0pt
}
{
\includegraphics[width=0.245\textwidth]{NoUVB_vs_Heat_z21.Species.eps}
\includegraphics[width=0.245\textwidth]{NoUVB_vs_Heat_z19.Species.eps}
\includegraphics[width=0.245\textwidth]{NoUVB_vs_Heat_z17.Species.eps}
\includegraphics[width=0.245\textwidth]{NoUVB_vs_Heat_z14.Species.eps}
}
 \caption{
The same quantities as in Fig. \ref{fig:early_profiles}, but for a halo which experiences positive feedback in the Heat0.8\ run. Solid and dashed curves correspond to the NoUVB\ and Heat0.8\ runs, respectively.
\label{fig:NoUVB_vs_Heat}
}
\vspace{-1\baselineskip}
\end{figure*}

\begin{figure*}
\vspace{+0\baselineskip}
{
\includegraphics[width=0.245\textwidth]{LW3NoUVB_vs_LW3Heat_z21.eps}
\includegraphics[width=0.245\textwidth]{LW3NoUVB_vs_LW3Heat_z19.eps}
\includegraphics[width=0.245\textwidth]{LW3NoUVB_vs_LW3Heat_z17.eps}
\includegraphics[width=0.245\textwidth]{LW3NoUVB_vs_LW3Heat_z14.eps}
\vskip0.0pt
}
{
\includegraphics[width=0.245\textwidth]{LW3NoUVB_vs_LW3Heat_z21.Species.eps}
\includegraphics[width=0.245\textwidth]{LW3NoUVB_vs_LW3Heat_z19.Species.eps}
\includegraphics[width=0.245\textwidth]{LW3NoUVB_vs_LW3Heat_z17.Species.eps}
\includegraphics[width=0.245\textwidth]{LW3NoUVB_vs_LW3Heat_z14.Species.eps}
}
 \caption{
Same as Fig. \ref{fig:NoUVB_vs_Heat}, but with solid and dashed curves corresponding to the LW10$^{-3}$\_NoUVB\ and LW10$^{-3}$\_Heat0.8\ runs, respectively.
\label{fig:LW3NoUVB_vs_LW3Heat}
}
\vspace{-1\baselineskip}
\end{figure*}

How does adding a LWB impact this positive feedback?  From Fig. \ref{fig:LW_delta}, we have already surmised that a background intensity of $J_{21}^{\rm LW}\gsim10^{-3}$, can offset these trends by decreasing the enhanced H$_2$ fraction in relic HII regions.  In Fig. \ref{fig:LW3NoUVB_vs_LW3Heat}, we plot profiles from the same halo in the LW10$^{-3}$\_NoUVB\ ({\it solid curves}) and LW10$^{-3}$\_Heat0.8\ ({\it dashed curves}).  By comparing the solid curves from Fig. \ref{fig:NoUVB_vs_Heat} to the dashed curves from Fig. \ref{fig:LW3NoUVB_vs_LW3Heat}, one confirms  that a LWB with $J_{21}^{\rm LW}\sim10^{-3}$ effectively neutralizes the positive feedback from the transient UVB.   Thus at $z=14$ the halo core has a similar gas profile and cooling time in the NoUVB\ and LW10$^{-3}$\_Heat0.8\ runs.

By comparing the two profiles at $z=14$ in Fig. \ref{fig:LW3NoUVB_vs_LW3Heat}, we see that the transient UVB still stimulates positive feedback, but the effect is not nearly as dramatic.  Specifically, in the halo core at $z=14$ the density is enhanced and the cooling time is lowered (both by a factor of $\sim2$) in the LW10$^{-3}$\_Heat0.8\ run compared to the LW10$^{-3}$\_NoUVB\ run.  This level of positive feedback is much more modest compared with the case with no LWB, where the equivalent density enhancement was a factor of $\sim50$ and the cooling time was lower by a factor of $\sim15$ (see Fig. \ref{fig:NoUVB_vs_Heat}).

Does this trend continue as we increase the strength of the UVB?  In Fig. \ref{fig:LW2NoUVB_vs_LW2Heat}, the solid and dashed curves correspond to the LW10$^{-2}$\_NoUVB\ and LW10$^{-2}$\_Heat0.8\ runs at $z=14$, respectively; similarly in Fig. \ref{fig:LW1NoUVB_vs_LW1Heat}, the solid and dashed curves correspond to the LW0.1\_NoUVB\ and LW0.1\_Heat0.8\ runs at $z=14$, respectively.  Here we see that the core density in the LW0.1\_Heat0.8\ run is lower than in the LW0.1\_NoUVB\ run, but the cooling time is {\it also lower} by the same factor of $\sim2$ seen in Fig. \ref{fig:LW3NoUVB_vs_LW3Heat}, sourced by the enhanced H$_2$ fraction.  In fact, from higher redshift output and the previous figures, we see that the density is still ``catching up'' as time progresses; namely, the dashed density curve keeps getting closer to the solid density curve.  Thus we see that {\it halos inside relic HII regions, when exposed to the same LWB, always have a higher H$_2$ abundance and shorter cooling times than halos outside relic HII regions, allowing gas to cool faster once it finally begins to collapse onto the halo}.

It is important to note that in this regime where a strong LWB dominates the overall feedback, star formation is likely to be substantially delayed.  This delay can be long enough to allow the halo to be photoevaporated during reionization, before it had a chance to host additional stars.  Thus if one is only concerned with the fate of those minihalos contributing their ionizing photons to the progress of reionization, it is possible that the sole {\it persistent} feedback mechanism is {\it positive}.

\begin{figure}
\includegraphics[width=0.45\textwidth]{LW2NoUVB_vs_LW2Heat_z14.eps}
\includegraphics[width=0.45\textwidth]{LW2NoUVB_vs_LW2Heat_z14.Species.eps}
 \caption{
Same as Fig. \ref{fig:NoUVB_vs_Heat}, at $z=14$, but with solid and dashed curves corresponding to the LW10$^{-2}$\_NoUVB\ and LW10$^{-2}$\_Heat0.8\ runs, respectively.
\label{fig:LW2NoUVB_vs_LW2Heat}
}
\vspace{-1\baselineskip}
\end{figure}

\begin{figure}
\includegraphics[width=0.45\textwidth]{LW1NoUVB_vs_LW1Heat_z14.eps}
\includegraphics[width=0.45\textwidth]{LW1NoUVB_vs_LW1Heat_z14.Species.eps}
 \caption{
Same as Fig. \ref{fig:NoUVB_vs_Heat}, at $z=14$, but with solid and dashed curves corresponding to the LW0.1\_NoUVB\ and LW0.1\_Heat0.8\ runs, respectively.
\label{fig:LW1NoUVB_vs_LW1Heat}
}
\vspace{-1\baselineskip}
\end{figure}

\section{The Impact of Neglecting Radiative Transfer}
\label{sec:ss}

As we have noted a number of times in this paper, our simulations assume the optically thin limit, for both ionizing radiation and photodissociating LW photons.  Both of these approximations clearly fail at points in our simulations, and so it is necessary to understand the nature and implications of this approximation.  For clarity, we discuss ionizing photons and LW photons separately.

We discuss the impact of radiation hydrodynamic (RHD) effects due to ionizing photons first.  There is now a substantial body of literature on the ionization of primordial gas halos \citep{SIR04, WAN04, ISR05, ABS05, SU06, Whalen08, AWB07, WA08} and the broad outline of how halos are ionized is now well understood.  We begin by noting that the optical depth of the halos which are ionized in our simulation box at $z=25$ (as shown in Fig.~\ref{fig:early_profiles}), as measured from the virial radius to the center is -- at the Lyman limit -- well over 100.  Therefore, at first blush, the optically thin limit seems like a poor approximation.  However, this impression is not entirely accurate for two reasons.

First, we are studying the {\em later} formation of halos in relic HII regions, and much of the gas that eventually falls into a halo at late times is {\em not} in the center of halos at reionization.  This is the question that \citet{OH03} asked: does the excess entropy generated in low-density gas in relic HII regions prevent it from forming stars at later times.  This low density gas has much lower optical depth and so is very likely to be ionized by a nearby Pop III star.

Second, even for the gas in the centers of our halos at $z_{\rm UVB, on}$, the optically thin approximation is not as bad as the large optical depth at the Lyman limit indicates.  We can see this more clearly by comparing our results to RHD calculations.  For example, take the (typical) halo shown in Fig.~\ref{fig:early_profiles}.  The central density when ionization begins (at $z=25$) is just slightly over 1 cm$^{-3}$.  This value is much below the central density ($10^4$ cm$^{-3}$) assumed by \citet{SU04}, and is not that different from their critical density for self shielding (depending on the assumed distance of the source).  A better comparison is with the ``023" series of simulations run in \citet{Whalen08}, which have almost identical central densities.  As is shown in Fig. 17 of that paper, this halo is disrupted and dissociated for all assumed distances of the first star.  Comparing Fig. 9 and 10 of that paper to our Fig.~\ref{fig:early_profiles} shows some obvious differences due to radiative transfer effects, but the basic evolution is not all that different.  In particular, after a few Myr, the density at the halo center has decreased by about a factor of $\sim 5$ after $\sim 3$ Myr, while the molecular hydrogen fraction has increased to a few times $10^{-3}$.  These numbers are very similar to what we find, and it is these basic facts which, we argue, cause the delay in future star formation, as discussed earlier.  The reason for the similarity is that, as it sweeps over the halo, the ionization front can quickly ionize hydrogen until it transitions to a D-type front and stalls.  However, this happens only $\sim 15$ pc from the core, at densities slightly below 1 cm$^{-3}$, and the shock generated by the D-type front quickly disperses the core.

Of course, we do not claim to reproduce the radiation hydrodynamics in detail.  For example, we do not reproduce the thin self-shielded filament at the lower right of Fig. 10 of \citet{Whalen08}, nor do we follow the detailed shock-compression or "rocket" offset of the core.   However, by choosing to apply the ionizing flux at a redshift when the halos have low masses and central densities, we minimize the impact of the radiation transfer.  To verify that this is actually true in our simulations, we examined the central densities and masses of our halos at $z=25$; we find that all halos except for one have masses below $10^6$ $M_\odot$ and central densities below $\sim$ 2 cm$^{-3}$ (one halo has $M > 10^6 M_\odot$ and $n_c = 7 \times 10^3$ cm$^{-3}$).  Our halo abundance is consistent with predictions from the conditional Press-Schechter formalism (MBH06).  We chose this approach because we are interested in investigating the impact of relic HII regions on the formation of structure which forms significantly later, rather than studying the impact of a Pop III star on a nearby, highly collapsed halo.  In particular, as noted in the introduction, our motivation is to follow up on the suggestion of \citet{OH03} that relic HII regions will imprint an excess entropy on the low-density gas at early times, which will then suppress later star formation.  Of course, we stress that it is still important to examine our results with a more realistic RHD treatment.

The results discussed in the section on positive feedback (section~\ref{sec:pos}) are even more robust because in that case the halos form out of gas that was barely overdense when the ionization occurred.  As can be seen in Fig.~\ref{fig:NoUVB_vs_Heat}, even at $z=21$ (long after the transient ionization), the central density of the halo (in the noUVB case) is only 0.2 cm$^{-3}$.

Another useful comparison of our results can be made with \citet{WA08b}, who used a full cosmological simulation of Pop III star formation including radiative transfer (albeit in a region considerably smaller than considered there).   While a direct comparison is difficult because \citet{WA08b} do not explicitly measure the suppression due to photo-ionization, a number of conclusions can be drawn.  First, it is clear from Fig. 7 of that paper that they also see the evaporation of gas from low-mass halos and filaments, and the resulting density field in the two simulations looks remarkably similar.  In addition, the time difference between the first Pop III object and the formation of the second is about 80 Myr, consistent with the time-delay derived in this paper.  On the other hand, one important point that comes out of that paper is that, because Pop III star formation tends to be clustered, relic HII regions do not sit isolated for longer periods of time and so a region of space may undergo multiple ionization episodes, something which is not considered in this work.

Next, we turn to LW photons.  First, we remind the reader that we are discussing a persistent LW background, and that although the LW flux is produced by the same {\it class} of objects that generate the transient UV flux, it is not produced by the same physical objects.  As discussed in more detail in \citet{HRL96}, the LW background is accumulated from a large number of distant sources.
The effective optical depth in the LW lines in the background IGM,
over a Hubble distance, is at most $\tau\approx 1-2$
\citep{HAR00,RGS01}, so the Olbers' integral for the LWB is dominated
by the large number of sources at a good fraction of the Hubble
distance.  In comparison, the ionizing photons come exclusively from
the much fewer source(s) nearby, within the local HII region.

LW photons can be self-shielded by a sufficient column of H$_2$.  Here again, there are two regimes.  One is the large quantities of low-density gas in the relic HII regions that form substantial amounts of molecular hydrogen immediately after the ionizing flux is turned off (e.g. \citealt{KM05}).  Although H$_2$ formed in this manner may slow the build-up of a persistent LW background, it does not -- for a given LW background -- provide a large amount of shielding.  For example, \citet{JGB07} find typical values of LW optical depths of order unity for a single relic HII region.   This means that the LW flux inside the relic HII region can be reduced by a factor of a few, but it also means that it doesn't take long for the persistent LW background to "burn off" the H$_2$ produced in relic HII regions.

The second regime occurs after the ionized gas falls back into the collapsed halos, where H$_2$ is in equilibrium between formation and photo-destruction.  In this case, when a sufficient column density of H$_2$ is formed, it can effectively attenuate the LW flux.  Based on our simulations, we find that when CD gas forms, the LW optical depth can be of order unity or larger, indicating that self-shielding can play a role during the recollapse (although probably not during the photoionization process itself).  Therefore, we caution readers that the critical LW values we give above may need to be increased when self-shielding is properly accounted for.

Finally, we note that in simulations with full radiative transfer, the photoionizing flux can also vary across a given halo (relic H II regions have radii of 2-3 kpc, while the Lagrangian radius of a typical halo in our simulations is of order 0.5 kpc).  This also means that, while most forming halos lie either completely inside or outside a relic H II region, some proto-halos could be partially ionized.

\section{Conclusions}
\label{sec:conc}

UV radiation from early astrophysical sources could have a large impact on subsequent star formation in nearby protogalaxies, and in general on the progress of cosmological reionization.  Theoretical arguments based on the absence of metals in the early Universe suggest that the first stars were likely massive, bright, yet short-lived, with lifetimes of a few million years.  Here we study the impact of the transient radiation arising from such stars on early protogalaxies.  We apply the same statistical approach as in MBH06, studying various combinations of transient UVBs and persistent LWBs, using the hydrodynamical simulation code, Enzo.  We also study a more typical and relevant region whose proto-galaxies form at lower redshifts, $z\sim$ 13--20. This allows us to trace feedback effects on longer time-scales and lower redshifts.

We confirm the results of MBH06 that feedback in the relic HII regions resulting from such transient radiation, is itself transient.  Feedback effects dwindle away after $\sim30\%$ of the Hubble time, and the same critical specific intensity of $J_{\rm UV} \sim 0.1 \times 10^{-21}{\rm ergs~s^{-1}~cm^{-2}~Hz^{-1}~sr^{-1}}$ separates positive and negative feedback regimes. A weaker UVB stimulates subsequent star formation
inside the relic HII regions, by enhancing the molecular hydrogen abundance. A stronger UVB delays star-formation by reducing the gas density at the centers of collapsing halos. The fact that we have confirmed the results in MBH06 (which were mostly inferences obtained by extrapolating from higher redshifts since we were unable to continue those simulations to low enough redshifts) suggests that overall feedback is fairly  insensitive to the large-scale environment, overdensity, and redshift-dependent halo parameters, and can accurately be modeled in this regime with just the intensity of the impinging UVB.

 We note that a value of $J_{21}^{\rm LW}\sim 10^{-3}$--$10^{-2}$ separates feedback regimes dominated by the LW background from those dominated by the transient UVB.  As discussed in section~\ref{sec:ss}, there is some uncertainty in these values due to our optically thin treatment.  In particular, accounting for LW self-shielding may increase the values of $J_{21}^{\rm LW}$ quoted here.

Additionally, we discover a second episode of eventual positive feedback in halos which have not yet collapsed when their progenitor regions were exposed to the transient UVB.
Gas in such late-forming objects did not have time to set-up a steep density profile, which would translate into a strong pressure shock when photo-heated.  Instead they formed out of material which was of more moderate densities at $z_{\rm UVB, on}$ where the lasting impact of the transient UVB was positive: seeding the gas with an enhanced abundance of molecular hydrogen, which aids in its eventual cooling.
However, this feedback regime is very sensitive to the presence of Lyman-Werner radiation, and disappears under fairly modest background intensities of $J_{\rm LW} \gsim 10^{-3} \times 10^{-21}{\rm ergs~s^{-1}~cm^{-2}~Hz^{-1}~sr^{-1}}$.
Nevertheless, when exposed to the same LWB, halos inside relic HII regions always have a higher H$_2$ abundance and shorter cooling times than halos outside relic HII regions, allowing gas to cool faster once it finally begins to collapse onto the halo.
  Although it is difficult to accurately estimate the build-up of the LWB, it seems likely that such modest values were surpassed well before the bulk of reionization \citep{HAR00}.  Thus this persistent eventual positive feedback might only be of academic interest, provided LW self-shielding is not efficient.

We conclude that radiative feedback from a short-lived UVB seems unlikely to have a major impact on the progress of global cosmological reionization, provided that present estimates of the luminosities and lifetimes of Pop III stars are accurate (e.g. \citealt{Schaerer02}).  Thus it is likely that the LWB plays the dominant role in regulating feedback in relic HII regions.

\vskip+0.5in

We thank the referee for comments which significantly improved the presentation of this paper.  Support for this work was provided by NASA through Hubble Fellowship grant \#HF-01222.01 to AM, awarded by the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Inc., for NASA, under contract NAS 5-26555.  GB acknowledges support from NSF grants AST-05-07161, AST-05-47823, and AST-06-06959, as well as computational resources from the National Center for Supercomputing Applications.
ZH thanks the Pol\'{a}nyi Program of the Hungarian National Office of Technology.

\bibliographystyle{mn2e}


\section{Introduction}
Since the beginning of functional analysis, a lot of questions have been
asked about the possible structure of Banach spaces, and
it turned out that one can construct spaces having the strangest properties.
Among the most prominent examples are the reflexive Tsirelson space
into which neither an $\ell^p$ space nor $c_0$ can be embedded~\cite{Tsi74},
a space found by Gowers and Maurey not containing an unconditional basic sequence~\cite{GM93},
and the recent positive solution to the
``scalar plus compact problem'' due to S.\ Argyros and R.\ Haydon.
A more detailed list of exotic properties that Banach spaces may exhibit
can be found in the survey articles of Maurey~\cite[Chapter~29]{JL01}
and Gowers~\cite{Gow94}.

Despite or maybe because of the abundance of vastly different Banach spaces
people started to classify the spaces on the basis of some of their most
important characteristics in order
to find some general principles how the various kinds of Banach spaces look like.
Simple criteria to distinguish spaces include qualitative properties such as
reflexivity or separability. More sophisticated investigations also aim towards
qualitative descriptions of the structure, for
example the celebrated Rademacher type and cotype which was first investigated
by Maurey and Pisier in the 1970s.
Soon it turned out that the type and cotype already carry enough information to
characterize the Hilbert spaces among all Banach spaces;
this is a deep result due to Kwapie\'n~\cite{Kwapien72}.
It should be pointed out that the Rademacher type is a local concept, i.e.,
its numerical value for a normed space $X$ depends only on the structure of
the finite dimensional subspaces of $X$.

In the present article we propose a new qualitative measure of the geometry of Banach spaces,
i.e., a new kind of type, which we call the \emph{Banach-Mazur type}.
This is a global concept which was implicitly introduced in Banach's book~\cite[XII.\S 3]{Banach32}
as joint work with Mazur for the special case of $L^p(0,1)$ and $\ell^p$ in order
to investigate the so-called \emph{linear dimension} of those spaces, see
Section~\ref{lineardimension}.
Although it is possible to determine the Banach-Mazur type
of $\ell^p$ by elementary calculations similar for example to the techniques of Banach's
proof of Schur's theorem~\cite[p.~137]{Banach32}, we follow a more elegant approach
based on properties of the canonical Schauder basis of $\ell^p$.
As an immediate application
we are able to prove the well-known fact that $\ell^p$ is not isomorphic to $\ell^q$
for $1 < p < q < \infty$, compare Remark~\ref{ellpellqnoniso}.
Another interesting point is that in contrast to the type and cotype introduced by Maurey and Pisier
the Banach-Mazur type is able to show that for any $p \in (2,\infty)$
the spaces $\ell^p$ and $L^p(0,1)$ are not isomorphic.

The article is organized as follows.
In Section~\ref{definition} we define the Banach-Mazur type and the unconditional
Banach-Mazur type. In Section~\ref{properties} we collect some properties which prove to
be useful for the calculation of the type.
Section~\ref{lineardimension} generalizes some ideas of~\cite[XII.\S 3]{Banach32} and shows how this
technique can be used to investigate the so-called \emph{linear dimension} of Banach spaces.
Section~\ref{sums} relates the unconditional Banach-Mazur type of a product of Banach spaces to
the unconditional Banach-Mazur type of its factors. This is interesting mainly because it provides us
with an example where the unconditional Banach-Mazur type is more useful than the Banach-Mazur type.
To actually compute the type of classical spaces, we will utilize basic facts from
the theory of Schauder bases. For the abstract setting this is done in Chapter~\ref{schauderbases}.
Finally, Section~\ref{examples} explicitly determines the Banach-Mazur types for
some classical Banach spaces by applying the results of the preceding sections.
Of course, a lot of interesting problems remain open after this rough investigation
of the Banach-Mazur type. Some questions are collected in Section~\ref{openproblems}.

\section{Definition}\label{definition}
\begin{definition}\label{maindef}
	Let $X$ be a normed vector space, $p \in [1,\infty]$.
	As usual, we set $\frac{1}{\infty} \colonequals 0$.
	\begin{enumerate}[(i)]
	\item
		We say that
		$X$ is \emph{of Banach-Mazur type $p$} (or simply \emph{of type $p$})
		if the following condition is fulfilled.
		Whenever $(x_n)$ is a sequence in $X$ which converges weakly to $0$
		there exists a subsequence $(x_{n_k})$ of $(x_n)$ and a constant
		$C > 0$ such that $\bigl\| \sum_{k=1}^N x_{n_k} \bigr\| \le C N^{\nicefrac{1}{p}}$
		for all $N \in \mathds{N}$.
	\item\label{maindef:uncond}
		We say that $X$ is \emph{of unconditional Banach-Mazur type $p$}
		(or simply \emph{of unconditional type $p$}) if the following condition
		is fulfilled.
		Whenever $(x_n)$ is a sequence in $X$ which converges weakly to $0$
		there exists a subsequence $(x_{n_k})$ of $(x_n)$ such that for
		every subsequence $(x_{n_{k_\ell}})$ of $(x_{n_k})$ we find a constant $C>0$
		satisfying $\bigl\| \sum_{\ell=1}^N x_{n_{k_\ell}} \bigr\| \le C N^{\nicefrac{1}{p}}$
		for all $N \in \mathds{N}$.
	\item
		We call
		\begin{equation}\label{def:type}
			\Type(X) \colonequals \sup \left\{ p \in [1,\infty] : X \text{ is of type $p$} \right\}
		\end{equation}
		the \emph{(Banach-Mazur) type of $X$} and
		\begin{equation}\label{def:utype}
			\UType(X) \colonequals \sup \left\{ p \in [1,\infty] : X \text{ is of unconditional type $p$} \right\}
		\end{equation}
		the \emph{unconditional (Banach-Mazur) type of $X$}.
	\end{enumerate}
\end{definition}

\section{Basic Properties}\label{properties}

Some properties of the type and the unconditional type are immediately obvious.
\begin{enumerate}[(P1)]
\item
	Because every weakly convergent sequence is bounded,
	the sets in~\eqref{def:type} and \eqref{def:utype} are not empty, i.e.,
	$\Type(X) \ge 1$ and $\UType(X) \ge 1$.
\item\label{interval}
	If $X$ is of type $p$, then $X$ is also of type $q$ whenever $1 \le q \le p$,
	and similarly for the unconditional type.
	Thus the sets in~\eqref{def:type} and~\eqref{def:utype} are in
	fact intervals.
\item\label{untypetotype}
	If $X$ is of unconditional type $p$, then $X$ is also of type $p$,
	hence $\UType(X) \le \Type(X)$ for every normed space $X$.
\end{enumerate}

Now we compare the type of spaces which are related by inclusion or isomorphy.
\begin{lemma}\label{monotonicity}
	Let $X$ be a normed space of type $p \in [1,\infty]$ and let $U$ be a subspace of $X$
	equipped with the induced norm. Then also $U$ is of type $p$.
	A similar statement is true for the unconditional type.
	Hence $\Type(U) \ge \Type(X)$ and $\UType(U) \ge \UType(X)$.
\end{lemma}
\begin{proof}
	Let $X$ be of type $p$ and
	let $(x_n)$ be a sequence in $U$ which converges weakly to $0$ in $U$.
	Then $(x_n)$ can also be considered as a sequence in $X$, and it
	is easily checked that it converges to $0$ weakly in $X$.
	Hence by definition $\bigl\| \sum_{k=1}^N x_{n_k} \bigr\| \le C N^{\nicefrac{1}{p}}$
	for some subsequence and some constant $C$.
	The same argument applies to the unconditional type.
\end{proof}

\begin{lemma}\label{completion}
	Let $X$ be a normed space and let $Y$ be
	its completion. Then $X$ is of type $p$ if and
	only if $Y$ is of type $p$.
	The same is true for the unconditional type.
	Thus $\Type(X) = \Type(Y)$ and $\UType(X) = \UType(Y)$.
\end{lemma}
\begin{proof}
	Because $X$ is a subspace of $Y$ one of the implications follows from Lemma~\ref{monotonicity}.
	Now let $X$ be of type $p$. Fix a sequence $(y_n)$ in $Y$ such that
	$y_n \rightharpoonup 0$ in $Y$. Since $X$ is dense in $Y$ there exist vectors
	$x_n \in X$ such that $\| x_n - y_n \| < \frac{1}{2^n}$.
	Note that then also $x_n \rightharpoonup 0$ in $X$. Hence by assumption there exist a
	subsequence $(x_{n_k})$ and a constant $C$
	such that $\bigl\| \sum_{k=1}^N x_{n_k} \bigr\| \le C N^{\nicefrac{1}{p}}$.
	For the corresponding subsequence $(y_{n_k})$ we have
	\[
		\Bigl\| \sum_{k=1}^N y_{n_k} \Bigr\|
			\le \Bigl\| \sum_{k=1}^N x_{n_k} \Bigr\| + \sum_{k=1}^N \bigl\| x_{n_k} - y_{n_k} \bigr\|
			\le C N^{\nicefrac{1}{p}} + 1
			\le (C+1) N^{\nicefrac{1}{p}}.
	\]
	This shows that $Y$ is of type $p$.
	The same proof works for the unconditional type.
\end{proof}

The Banach-Mazur type is non-local, i.e., it can not be computed from
its values for the finite dimensional subspaces.
However, it depends only on the structure of the separable subspaces.
We record this observation in the following lemma.
\begin{lemma}\label{separable}
	Let $X$ be a normed space. Then
	\[
		\Type(X) = \inf_{U} \Type(U)
		\quad\text{and}\quad
		\UType(X) = \inf_{U} \UType(U),
	\]
	where $U$ varies over all closed separable subspaces of $X$.
\end{lemma}

\begin{proof}
	It follows from Lemma~\ref{monotonicity} that $\Type(X) \le \inf_{U} \Type(U)$.
	Hence for $T(X) = \infty$ the first equality is proved. Otherwise, i.e., if $T(X) < \infty$,
	let $q > \Type(X)$ be arbitrary. Then there exists a sequence
	$(x_n)$ in $X$ such that $x_n \rightharpoonup 0$ in $X$ and
	\[
		\sup_{N \in \mathds{N}} \frac{\bigl\| \sum_{k=1}^N x_{n_k} \bigr\|}{N^{\nicefrac{1}{q}}} = \infty
	\]
	for every subsequence $(x_{n_k})$ of $(x_n)$.
	Denote by $V$ the space closure of the space $\hull\{ x_n : n \in \mathds{N}\}$.
	Then $V$ is a closed, separable subspace of $X$ and
	it follows from the Hahn-Banach theorem that $x_n \rightharpoonup 0$ in $V$.
	By construction $V$ is not of type $q$, hence $\Type(V) \le q$
	according to (P\ref{interval}). This shows
	$\inf_U \Type(U) \le \Type(V) \le q$. Taking the infimum over all $q > T(X)$
	we obtain $\inf_U \Type(U) \le \Type(X)$.
	The proof for the unconditional type is similar.
\end{proof}

\begin{lemma}\label{isomorphyinvariance}
	If two normed spaces $X$ and $Y$ are isomorphic, then $X$ is of
	type $p$ if and only if $Y$ is of type $p$.
	A similar statement holds for the unconditional type.
	In particular $\Type(X) = \Type(Y)$ and $\UType(X) = \UType(Y)$.
\end{lemma}

\begin{proof}
	Let $X$ be of type $p$ and let $J\colon X \to Y$ be an isomorphism.
	Fix a sequence $(y_n)$ satisfying $y_n \rightharpoonup 0$. Then also
	$x_n \colonequals J^{-1} y_n \rightharpoonup 0$. Hence there exists a subsequence
	$(x_{n_k})$ and a constant $C$ such that
	$\bigl\| \sum_{k=1}^N x_{n_k} \bigr\| \le C N^{\nicefrac{1}{p}}$.
	Consequently, for the corresponding subsequence $y_{n_k} = Jx_{n_k}$,
	$\bigl\| \sum_{k=1}^N y_{n_k} \bigr\| \le \|J\| C N^{\nicefrac{1}{p}}$.
	This shows that also $Y$ is of type $p$.
	Since the statement is symmetric with respect to $X$ and $Y$, the claim
	follows for the type. The proof for the unconditional type is
	almost identical.
\end{proof}

\section{Applications to the Linear Dimension}\label{lineardimension}

Banach and Mazur introduced the Banach-Mazur type only for the spaces $\ell^p$ and $L^p$.
They used it as a tool to compare the so-called \emph{linear dimension} of these spaces.
Their ideas can be applied in an abstract situation. As a special case we obtain
their original results. We show Corollary~\ref{incomplindim} as an example,
but the other results about the linear dimensions of $\ell^p$ and $L^p(0,1)$, $p \in (1,\infty)$,
can be obtained in the same way.

Let $X$ and $Y$ be two Banach spaces. We say that \emph{the linear dimension of $X$ does
not exceed the linear dimension of $Y$} if $X$ is isomorphic to a closed subspace of $Y$,
and for this we write $\dim_\ell X \le \dim_\ell Y$.
If neither $\dim_\ell X \le \dim_\ell Y$ nor $\dim_\ell Y \le \dim_\ell X$ holds, we say that
\emph{$X$ and $Y$ are of incomparable linear dimension}.

\begin{lemma}\label{quotienttype}
	Let $X$ and $Y$ be Banach spaces, let $X$ be reflexive, and let $T \in \mathscr{L}(X,Y)$ be onto.
	If $X$ is of type $p \in [1,\infty]$, then also $Y$ is of type $p$.
	A similar statement is true for the unconditional type.
	Hence $\Type(X) \le \Type(Y)$ and $\UType(X) \le \UType(Y)$.
\end{lemma}
\begin{proof}
	The operator $T$ is open according to the open mapping theorem. Hence there exists
	$c > 0$ such that for every $y \in Y$ we find $x \in X$ satisfying $Tx = y$ and
	$\|x\| \le c\|y\|$. Let $(y_n)$ be a sequence in $Y$, $y_n \rightharpoonup 0$, and choose $(x_n)$
	such that $Tx_n = y_n$ and $\|x_n\| \le c\|y_n\|$. Then $(x_n)$ is bounded in $X$.
	After passing to a subsequence we may therefore assume that $x_n \rightharpoonup x$.
	Then $y_n = Tx_n \rightharpoonup Tx$, thus $Tx = 0$.
	Letting $\tilde{x}_n \colonequals x_n - x$ we have $\tilde{x}_n \rightharpoonup 0$ and
	$T\tilde{x}_n = y_n$. By assumption we can choose a subsequence of $(\tilde{x}_n)$ such that
	$\bigl\| \sum_{k=1}^N \tilde{x}_{n_k} \bigr\| \le C_X N^{\nicefrac{1}{p}}$ for some constant
	$C_X > 0$. But then $\bigl\| \sum_{k=1}^N y_{n_k} \bigr\| \le C_Y N^{\nicefrac{1}{p}}$
	for $C_Y \colonequals \|T\| C_X$. This shows that also $Y$ is of type $p$.
	The same reasoning works for the unconditional type.
\end{proof}
\begin{remark}
	In particular, the lemma shows the following:
	Let $Y$ be a quotient of a reflexive Banach space $X$, i.e.,
	$Y$ is isometric to $X/Z$ for a closed subspace $Z$ of $X$.
	Then $\Type(X) \le \Type(Y)$ and $\UType(X) \le \UType(Y)$.
	This is similar to Lemma~\ref{monotonicity}.

	However, the condition of $X$ being reflexive cannot be dropped.
	To see this, recall that every separable Banach space is a quotient
	of $\ell^1$~\cite[Section~1.4]{JL01}, but $\Type(\ell^1) = \infty > 1 = \Type(C[0,1])$,
	see Examples~\ref{ell1} and~\ref{C01}.
\end{remark}

\begin{theorem}\label{dualtype}
	Let $X$ and $Y$ be Banach spaces and $\dim_\ell X \le \dim_\ell Y$.
	If $Y$ is of type $p \in [1,\infty]$, then also $X$ is of type $p$,
	thus $\Type(X) \ge \Type(Y)$.
	If $Y$ is reflexive and $Y'$ is of type $q \in [1,\infty]$, then
	also $X'$ is of type $q$, hence $\Type(X') \ge \Type(Y')$.
	Similar statements hold for the unconditional type.
\end{theorem}
\begin{proof}
	The first assertion follows from Lemmata~\ref{isomorphyinvariance} and~\ref{monotonicity}.
	Now let $Y$ (and hence also $Y'$) be reflexive and let $T\colon X \to Y$ be an isomorphism
	from $X$ onto a closed subspace $Z$ of $Y$.
	Then the adjoint $T'\colon Y' \to X'$ is onto. In fact, for $x' \in X'$ define
	$z' \colonequals x' \circ T^{-1} \in Z'$ and pick an
	extension $y' \in Y'$ of $z'$ which exists according to the Hahn-Banach theorem.
	Then $T'y' = x'$.
	By Lemma~\ref{quotienttype} this proves the second assertion.
\end{proof}

\begin{corollary}[\mbox{\cite[Th\'eor\`eme~XII.7]{Banach32}}]\label{incomplindim}
	The spaces $\ell^p$ and $\ell^q$
	are of incomparable linear dimension for $p,q \in (1,\infty)$, $p \neq q$.
\end{corollary}
\begin{proof}
	Assume $\dim_\ell \ell^p \le \dim_\ell \ell^q$ for some $p, q \in (1,\infty)$.
	According to Theorem~\ref{dualtype} this implies $\Type(\ell^p) \ge \Type(\ell^q)$
	and $\Type(\ell^{p'}) \ge \Type(\ell^{q'})$. Here $p'$ and $q'$ are the conjugate
	exponents of $p$ and $q$, respectively. By Example~\ref{ellp} this shows
	$p \ge q$ and $\frac{p}{p-1} \ge \frac{q}{q-1}$, implying $p=q$.
\end{proof}

\section{Direct Sums of Banach Spaces}\label{sums}
\begin{proposition}\label{spacesum}
	Let $X_i$ be normed spaces.
	The product $\bigoplus_{i=1}^N X_i$ is of unconditional type $p$
	if and only if all the spaces $X_i$ are of unconditional type $p$.
	In particular, we the unconditional type of the product space is
	$\UType(\bigoplus_{i=1}^N X_i) = \min_i \UType(X_i)$.
\end{proposition}
\begin{proof}
	Without loss of generality we prove the claim only for $N=2$.
	According to Lemma~\ref{isomorphyinvariance} we may choose
	the norm $\| (x, y) \| \colonequals \| x \|_{X_1} + \| y \|_{X_2}$
	on $X_1 \oplus X_2$. First let $X_1$ and $X_2$ be of unconditional type $p$.
	Let $(x_n, y_n) \rightharpoonup 0$ in $X_1 \oplus X_2$. Then $x_n \rightharpoonup 0$ in $X_1$
	and $y_n \rightharpoonup 0$ in $X_2$.
	We pick subsequences $(x_{n_k})$ of $(x_n)$ and $(y_{n_k})$ of $(y_n)$
	which simultaneously satisfy the growth conditions as in the definition
	of the unconditional type. Then for any subsequence we can estimate
	\[
		\Bigl\| \sum_{\ell=1}^N \bigl( x_{n_{k_\ell}}, y_{n_{k_\ell}} \bigr) \Bigr\|
			= \Bigl\| \sum_{\ell=1}^N x_{n_{k_\ell}} \Bigr\| + \Bigl\| \sum_{\ell=1}^N y_{n_{k_\ell}} \Bigr\|
			\le 2C N^{\nicefrac{1}{p}}.
	\]
	Thus $X_1 \oplus X_2$ is of unconditional type $p$.

	If, on the other hand, $X_1 \oplus X_2$ are of unconditional type $p$,
	then also its subspaces $X_1$ and $X_2$ are of unconditional type $p$
	according to Lemma~\ref{monotonicity}.
\end{proof}

\begin{remark}
	The above argument does not work for the type because it is not obvious whether we can pick subsequences
	which simultaneously satisfy the growth conditions. However, if $X_1$ is of
	type $p_1$ and $X_2$ is of unconditional type $p_2$, then
	$X_1 \oplus X_2$ is of type $\min\{ p_1, p_2 \}$, but not of type $q$ for any
	$q > \min\{ p_1, p_2 \}$. The proof is essentially the same as the above.
\end{remark}

\begin{proposition}
	Let $X_i$ be normed subspaces of unconditional type $p$ which are contained in
	a common vector space $Z$. Then the normed space
	$U \colonequals \bigcap_{i=1}^N X_i$ equipped with the usual norm
	$\| x \|_U \colonequals \sum_{i=1}^N \| x \|_{X_i}$ is of unconditional
	type $p$. In particular, $\UType(\bigcap_i X_i) \ge \min_i \UType(X_i)$.
\end{proposition}
\begin{proof}
	The space $\bigoplus_i X_i$ is of unconditional type $p$ by Proposition~\ref{spacesum}.
	Since $\bigcap_i X_i$ can be identified with the diagonal in $\bigoplus_i X_i$,
	Lemma~\ref{monotonicity} shows that also $\bigcap_i X_i$ is of unconditional type $p$.
\end{proof}

\begin{remark}
	The above estimate can in general not be improved.
	In fact, if $\bigcap_i X_i = \{0\}$,
	then $\UType(\bigcap_i X_i) = \infty$ independently of the unconditional types of the $X_i$.
\end{remark}

\section{Some Relations to Schauder bases}\label{schauderbases}
The following proposition can be applied to calculate the type of a whole class of Banach spaces.
We will use it to determine the type of $\ell^p$ and $c_0$.
Recall that a sequence $(x_n)$ in a Banach space $X$ is called a \emph{Schauder basis}
of $X$ if every $x \in X$ has a unique representation as a convergent series
$x = \sum_i \alpha_i x_i$. A sequence $(y_n)$ is called a \emph{basic sequence}
if it is a Schauder basis of $\overline{\hull}\{ y_n : n \in \mathds{N} \}$.
A basic sequence $(y_n)$ is called a \emph{block basic sequence of the basic sequence $(y_n)$}
if it can be written in the form $u_n = \sum_{i=m_n}^{m_{n+1}-1} \alpha_i y_i$ with a strictly increasing
sequence $(m_n)$ of natural numbers.
We call two basic sequences $(x_n)$ and $(y_n)$ \emph{equivalent} if
$Sx_n = y_n$ extends to an isomorphism between the spans of the sequences.

\begin{proposition}\label{blockbasis}
	Let $p \in [1,\infty]$ and $X$ be a Banach space.
	Let $(e_n)$ be a Schauder basis and assume that for every bounded block basic sequence
	$(y_\ell)$ of $(e_n)$ there exists a constant $c$ such that
	$\bigl\| \sum_{\ell=1}^N y_\ell \bigr\| \le c N^{\nicefrac{1}{p}}$ for all $N$.
	Then $X$ is of unconditional type $p$.
\end{proposition}

\begin{proof}
	Let $(x_n)$ be a sequence in $X$ such that $x_n \rightharpoonup 0$. We have to show
	that there exists a subsequence $(x_{n_k})$ such that every subsequence
	$(x_{n_{k_\ell}})$ of $(x_{n_k})$ we find a constant $C>0$ satisfying
	\begin{equation}\label{desestimate}
		\Bigl\| \sum_{\ell=1}^N x_{n_{k_\ell}} \Bigr\| \le C N^{\nicefrac{1}{p}} \text{ for all } N \in \mathds{N}.
	\end{equation}
	This is trivial if $\liminf_n \| x_n \| = 0$; in this case
	we may pick $C \colonequals 1$ for a subsequence $(x_{n_k})$ such that $\| x_{n_k} \| \le 2^{-k}$.
	So without loss of generality (passing to a subsequence if necessary) we may
	assume that there are constants such that $0 < \varepsilon_0 \le \| x_n \| \le M$ for all $n$.
	Now it is possible to pick a subsequence $(x_{n_k})$ of $(x_n)$ which is
	equivalent to a bounded block basic sequence $(y_k)$ of $(e_n)$,
	see~\cite[Proposition~1.a.12]{Sequences77}.
	In particular, the linear mapping $S\colon \hull\{ y_k \} \to X$ defined by
	$Sy_k \colonequals x_{n_k}$ is bounded.
	Let $(x_{n_{k_\ell}})$ be any subsequence of $(x_{n_k})$. Then $(x_{n_{k_\ell}})$
	is equivalent to the bounded block basic sequence $(y_{k_\ell})$ of $(e_n)$.
	From the assumption we then obtain \eqref{desestimate} for $C \colonequals c \left\|S\right\|$.
	This concludes the proof.
\end{proof}

The following lemma describes a situation in which we can assure that the type
and the unconditional type coincide.

\begin{proposition}\label{typeequncond}
	Let $p \in [1,\infty]$.
	Assume that $X$ is a Banach space with the following property.
	Every basic sequence $(x_n)$ in $X$ satisfying $x_n \rightharpoonup 0$ and $\liminf \| x_n \| > 0$
	possesses a subsymmetric subsequence $(x_{n_k})$, i.e.,
	every subsequence $(x_{n_{k_\ell}})$ of $(x_{n_k})$ is equivalent to $(x_{n_k})$.
	Then $X$ is of type $p$ if and only if $X$ is of unconditional type $p$.
\end{proposition}
\begin{proof}
	One implication is true in general, see (P\ref{untypetotype}).
	So now let $X$ be of type $p$ and let $(x_n)$ be such that $x_n \rightharpoonup 0$.
	As is the proof of Proposition~\ref{blockbasis} we may without loss of generality
	assume that $\liminf \| x_n \| > 0$.
	By passing to a subsequence if necessary we can achieve that $(x_n)$ is basic sequence,
	see the remarks following the proof of~\cite[Theorem~1.a.5]{Sequences77}.
	By assumption there exists a subsymmetric subsequence $(x_{n_k})$ of $(x_n)$.
	We claim that $(x_{n_k})$ has the property which Definition~\ref{maindef}.\eqref{maindef:uncond} requests.
	Since $X$ is of type $p$ we can find a subsequence $(x_{n_{\tilde{k}_\ell}})$ of $(x_{n_k})$
	and a constant $C > 0$ such that
	\begin{equation}\label{subsymmest}
		\Bigl\| \sum_{\ell=1}^N x_{n_{\tilde{k}_\ell}} \Bigr\| \le C N^{\nicefrac{1}{p}}
	\end{equation}
	holds for all $N \in \mathds{N}$.
	By subsymmetry $(x_{n_{\tilde{k}_\ell}})$ is equivalent to $(x_{n_k})$.
	Now if $(x_{n_{k_\ell}})$ is any subsequence of $(x_{n_k})$,
	by assumption $(x_{n_{k_\ell}})$ is equivalent to $(x_{n_k})$
	and hence also to $(x_{n_{\tilde{k}_\ell}})$.
	Thus, as in the proof of Proposition~\ref{blockbasis},
	\eqref{subsymmest} remains valid for $(x_{n_{k_\ell}})$ if we multiply
	$C$ by the operator norm of the corresponding isomorphism.
\end{proof}
\begin{example}
	The spaces $c_0$ and $\ell^p$, $1 \le p < \infty$ satisfy the assumptions
	of the Lemma~\ref{typeequncond}. To see this, let $(e_n)$ be the usual Schauder
	basis of the respective space and let $(x_n)$ be as in the lemma.
	We can find a subsequence $(x_{n_k})$ of $(x_n)$ which is equivalent to a block
	basic sequence of $(e_n)$, see~\cite[Proposition~1.a.12]{Sequences77}.
	Then $(x_{n_k})$ is equivalent to $(e_n)$~\cite[Proposition~2.a.1]{Sequences77}
	and hence subsymmetric.
\end{example}

\section{Examples}\label{examples}

First we consider a class of spaces whose type is particularly easy to compute.
A normed space $X$ is called a \emph{Schur space} if
every weakly convergent sequence is convergent in norm.
Examples of Schur spaces are finite dimensional spaces and $\ell^1$,
cf.~\cite[p.~137]{Banach32}.
\begin{example}\label{ell1}
	Every Schur space $X$ is of unconditional type $\infty$, hence
	$\UType(X) = \Type(X) = \infty$.
\end{example}
\begin{proof}
	Let $(x_n)$ be a sequence in $X$ such that $x_n \rightharpoonup 0$.
	Then $x_n \to 0$. Thus there exists a subsequence
	$(x_{n_k})$ such that $\| x_{n_k} \| \le 2^{-k}$.
	In particular, $\sum_{k=1}^N x_{n_{k_\ell}}$ is bounded in $X$
	for every subsequence $(x_{n_{k_\ell}})$ of $(x_{n_k})$.
	This shows that $X$ is of unconditional type $\infty$.
	Using (P\ref{untypetotype}), the claim is proved.
\end{proof}

\begin{example}\label{ellp}
	For every $p \in (1,\infty)$ the space $\ell^p$ is of unconditional type $p$,
	but not of type $q$ for any $q > p$, hence $\UType(\ell^p) = \Type(\ell^p) = p$.
	Moreover, $c_0$ is of unconditional type $\infty$, hence
	$\UType(c_0) = \Type(c_0) = \infty$.
\end{example}
\begin{proof}
	Consider the sequence $(e_n)$ of canonical unit vectors in $\ell^p$.
	It is easy to see that $e_n \rightharpoonup 0$ and that for any subsequence
	$(e_{n_k})$ we obtain
	\[
		\Bigl\| \sum_{k=1}^N e_{n_k} \Bigr\|_p
			= N^{\nicefrac{1}{p}}
		\quad \text{for all $N \in \mathds{N}$.}
	\]
	This shows that $\ell^p$ is not of type $q$ for any $q > p$.

	To show that $\ell^p$ is of unconditional type $p$ and that $c_0$
	is of unconditional type $\infty$ we apply Proposition~\ref{blockbasis}
	to the Schauder basis $(e_n)$ of canonical unit vectors.
	First consider the case of $\ell^p$.
	For any bounded block basic sequence $(y_\ell)$ of $(e_n)$ we obtain
	$\bigl\| \sum_{\ell=1}^N y_\ell \bigr\|_p^p = \sum_{\ell=1}^N \| y_\ell \|_p^p \le M^p N$
	by an easy calculation, where $M$ denotes an upper bound for $\|y_n\|_p$.
	Similarly, for a bounded block basic sequence $(y_\ell)$ in $c_0$ we have
	$\bigl\| \sum_{\ell=1}^N y_\ell \bigr\|_\infty = \max_\ell \| y_\ell \|_\infty \le M$,
	where $M$ denotes an upper bound for $\|y_n\|_\infty$.
	Now Proposition~\ref{blockbasis} asserts that $\ell^p$ is of unconditional type $p$ and
	$c_0$ is of unconditional type $\infty$.
	The remaining assertions follow from (P\ref{interval}) and (P\ref{untypetotype}).
\end{proof}

\begin{remark}\label{ellpellqnoniso}
	Example~\ref{ellp} provides a proof (via Lemma~\ref{isomorphyinvariance})
	of the fact that the sequence spaces $\ell^p$ and $\ell^q$
	are not isomorphic for $p \neq q$.
	Of course this is well-known and follows also from Pitt's Theorem~\cite[\S 42.3.(10)]{Koethe79} or
	the calculation of the Rademacher types and cotypes of these spaces~\cite{Schwartz81}.
\end{remark}

\begin{example}
	Let $I$ be an arbitrary infinite index set. We define $\ell^p(I)$ for $1 \le p < \infty$ as usual,
	i.e., $\ell^p(I)$ is the completion of the space of families $x = (x_\alpha)_{\alpha \in I}$
	where only finitely many components $x_\alpha$ do not vanish, with respect to the norm
	$\| x \|_p \colonequals \sum_{\alpha \in I} |x_\alpha|^p$.
	Then $\Type(\ell^p(I)) = \UType(\ell^p(I)) = p$.
	In particular, $\ell^p(I_1)$ and $\ell^q(I_2)$ are not isomorphic if $p \neq q$
	and either one of $I_1$ and $I_2$ is infinite.
\end{example}
\begin{proof}
	Since $\ell^p$ can naturally considered as a closed subspace of $\ell^p(I)$,
	Lemma~\ref{monotonicity} ensures that $\Type(\ell^p(I)) \le p$.
	If on the other hand $(x_n)$ is any sequence in $\ell^p(I)$ such that $x_n \rightharpoonup 0$ in $\ell^p(I)$,
	then the set $J$ of indices where at least one of the $x_n$ does not vanish, is at most countable.
	So $(x_n)$ is a sequence in the subspace $\ell^p(J)$ of $\ell^p(I)$ and $x_n \rightharpoonup 0$ in $\ell^p(J)$.
	Because $\ell^p(J)$ is isomorphic to $\ell^p$ or a finite dimensional space,
	the estimate on the partial sums follows either from Example~\ref{ell1} or Example~\ref{ellp}.
	This shows $\UType(\ell^p(I)) \ge p$.
\end{proof}

\begin{example}[Hilbert spaces]
	Every infinite dimensional inner product space $H$ is of unconditional type $2$,
	but not of type $q$ for any $q > 2$, hence $\UType(H) = \Type(H) = 2$.
\end{example}
\begin{proof}
	The case $H = \ell^2$ is a special case of Example~\ref{ellp}.
	Thus the claim holds for every infinite dimensional separable Hilbert space due to
	Lemma~\ref{isomorphyinvariance}.
	Now the assertion follows from Lemmata~\ref{completion} and~\ref{separable}.
\end{proof}

So far we have only seen examples of Banach spaces $X$ with $\Type(X) = \UType(X) = p$
for $p \in (1,\infty]$. The following example shows that also $p=1$ is possible.
\begin{example}\label{C01}
	$\Type(C[0,1]) = \UType(C[0,1]) = 1$.
\end{example}
\begin{proof}
	In fact, it is known that every separable Banach space $X$ is
	isometric to a closed subspace of $C[0,1]$~\cite[Section~1.4]{JL01}. Hence
	by Lemma~\ref{separable}
	\[
		\UType(C[0,1]) \le \Type(C[0,1]) \le \inf_{p \in (1,\infty)} \Type(\ell^p) = 1. \qedhere
	\]
\end{proof}

Recall that the Rademacher cotype of $\ell^p$ and $L^p(0,1)$ is $p$ for every $2 < p < \infty$.
For the Banach-Mazur type the situation is different as the following example shows.
\begin{example}\label{Lptype}
	Let $2 < p < \infty$. Then $\UType(L^p(0,1)) = \Type(L^p(0,1)) = 2$.
\end{example}
\begin{proof}
	It can be shown that $L^p(0,1)$ contains an isomorphic copy of $\ell^2$~\cite[Section~1.4]{JL01},
	so by Lemma~\ref{monotonicity} the type of $L^p(0,1)$ can be at most $2$.
	But on the other hand, if $(f_n)$ is a sequence in $L^p(0,1)$ such that
	$0 < \limsup_n \|f\|_n < \infty$, then $(f_n)$ has a subsequence which is equivalent
	to the unit vector sequence in either $\ell^2$ or $\ell^p$~\cite[Section~1.8]{JL01}. In both cases
	we obtain the estimate for the unconditional type, thus showing $\UType(L^p(0,1)) \ge 2$,
	and the assertion is proved.
\end{proof}
\begin{remark}
	It is possible to show that in fact $\Type(L^p(0,1)) = p$ for $p \in (1,2]$, see~\cite[Th\'eor\`eme~XII.2]{Banach32}.
\end{remark}
This shows that the spaces $\ell^p$ and $L^p(0,1)$ are non-isomorphic at least for $p \in (2,\infty)$
which does not follow from the value of the Rademacher types of these spaces.
Thus the Banach-Mazur type carries interesting information about a Banach space
which escapes some other concepts.

\section{Open Problems}\label{openproblems}
\begin{question}
	Does any Banach space $X$ satisfy $\UType(X) = \Type(X)$?
\end{question}
\begin{question}
	If $X$ is of type $p \colonequals \Type(X) < \infty$, is there a sequence $(x_n)$ in $X$
	such that any subsequence $(x_{n_k})$ of $(x_n)$ satisfies
	\[
		\limsup_{N \to \infty} \frac{\bigl\| \sum_{k=1}^N x_{n_k} \bigr\|}{N^{\nicefrac{1}{p}}} > 0,
	\]
	and similarly for the unconditional type? In other words, is there always a sequence
	for which the growth estimate is sharp?
\end{question}
\begin{question}
	Can the constant $C$ in~\eqref{def:type} and~\eqref{def:utype} always be chosen as
	\[
		C \le c_p \limsup_{n \to \infty} \| x_n \|,
	\]
	where $c_p$ is some universal constant depending only on $p$?
\end{question}

\section*{Acknowledgements}
The author expresses his gratitude to Prof.~Pietsch for valuable hints regarding the Banach-Mazur type,
in particular for pointing out that this concept has already been investigated
by Banach and Mazur.

The author thanks the Graduate School \textsl{Mathematical Analysis of Evolution, Information and Complexity}
for their support during the work on this article.

\bibliographystyle{amsalpha}