Statement:
stringlengths 7
24.3k
|
|---|
The function $r \mapsto \mathrm{of\_real}(r)$ is a bounded linear operator. |
A linear map $f$ from $\mathbb{R}$ to $\mathbb{R}$ is bounded if and only if there exists a real number $c$ such that $f(x) = cx$ for all $x \in \mathbb{R}$. |
In a metric space, the neighborhood filter of a point $x$ is the infimum of the principal filters generated by the open balls of radius $e$ centered at $x$, where $e$ ranges over the positive real numbers. |
A function $f$ tends to $l$ in a filter $F$ if and only if for every $\epsilon > 0$, the set $\{x \in X \mid |f(x) - l| < \epsilon\}$ is in $F$. |
$f$ converges to $l$ if and only if the distance between $f$ and $l$ converges to $0$. |
If $f$ is a function from a metric space to a metric space, and for every $\epsilon > 0$, there exists an $x$ such that $|f(x) - l| < \epsilon$, then $f$ converges to $l$. |
If $f$ converges to $l$ in a metric space, then for every $\epsilon > 0$, there exists an $x$ such that $|f(x) - l| < \epsilon$. |
In a metric space, a point $a$ has a neighborhood $U$ such that $P$ holds for all points in $U$ if and only if there exists a positive real number $d$ such that $P$ holds for all points $x$ with $d(x,a) < d$. |
A property $P$ holds eventually at $a$ within $S$ if and only if there exists a positive real number $d$ such that for all $x \in S$ with $x |
A property $P$ holds frequently at $a$ within $S$ if and only if for every $\epsilon > 0$, there exists $x \in S$ such that $x \neq a$ and $|x - a| < \epsilon$ and $P(x)$ holds. |
A property $P$ holds eventually at $a$ within $S$ if and only if there exists a positive real number $d$ such that for all $x \in S$ with $x \neq a$ and $d(x,a) \leq d$, $P(x)$ holds. |
If $a > b$, then eventually $x \in (b, a)$ as $x$ approaches $a$ from the left. |
If $a < b$, then there exists a neighborhood of $a$ such that all points in that neighborhood are in the interval $(a, b)$. |
If $f$ converges to $a$ and $g$ is eventually closer to $b$ than $f$ is to $a$, then $g$ converges to $b$. |
The sequence of real numbers converges to infinity. |
The natural numbers are cofinal in the set of all natural numbers. |
The sequence of floors of the natural numbers converges to infinity. |
A sequence $f$ converges to a filter $F$ if and only if the sequence of real numbers $f_n$ converges to $F$. |
A sequence $X$ converges to $L$ if and only if for every $\epsilon > 0$, there exists $N$ such that for all $n \geq N$, we have $|X_n - L| < \epsilon$. |
The limit of a sequence $a_n$ as $n$ goes to infinity is $L$ if and only if for every $\epsilon > 0$, there exists an $N$ such that for all $n > N$, we have $|a_n - L| < \epsilon$. |
A sequence $X$ converges to $L$ if and only if for every $\epsilon > 0$, there exists $N$ such that for all $n \geq N$, we have $|X_n - L| < \epsilon$. |
If for every $\epsilon > 0$, there exists $N$ such that for all $n \geq N$, we have $d(X_n, L) < \epsilon$, then $X_n$ converges to $L$. |
If $X_n$ converges to $L$, then for any $\epsilon > 0$, there exists $N$ such that for all $n \geq N$, we have $|X_n - L| < \epsilon$. |
If $f_n$ is a sequence of complex numbers such that $\|f_n\| < 1/n$ for all $n$, then $f_n \to 0$. |
A function $f$ converges to $L$ at $a$ if and only if for every $\epsilon > 0$, there exists $\delta > 0$ such that for all $x$, if $x \neq a$ and $|x - a| < \delta$, then $|f(x) - L| < \epsilon$. |
If for every $\epsilon > 0$, there exists $\delta > 0$ such that for all $x \in X$ with $x \neq a$ and $d(x, a) < \delta$, we have $d(f(x), L) < \epsilon$, then $f$ converges to $L$ at $a$. |
If $f$ converges to $L$ at $a$, then there exists a neighborhood of $a$ such that $f$ is within $r$ of $L$ on that neighborhood. |
If $f$ converges to $l$ and $g$ is closer to $m$ than $f$ is to $l$, then $g$ converges to $m$. |
Suppose $f$ and $g$ are real-valued functions defined on a metric space $X$, and $a$ is a limit point of $X$. If $g$ converges to $l$ at $a$, and $f$ and $g$ agree on the punctured ball of radius $R$ around $a$, then $f$ converges to $l$ at $a$. |
If $f$ and $g$ are continuous functions and $f$ is injective in a neighborhood of $a$, then $g \circ f$ is continuous at $a$. |
If $f$ is continuous at $a$ and $g$ is continuous at $f(a)$, then $g \circ f$ is continuous at $a$. |
A sequence $X$ in a metric space is Cauchy if and only if for every $\epsilon > 0$, there exists an $M$ such that for all $m, n \geq M$, we have $d(X_m, X_n) < \epsilon$. |
A sequence $f$ is Cauchy if and only if for every $\epsilon > 0$, there exists an $M$ such that for all $m \geq M$ and all $n > m$, we have $|f(m) - f(n)| < \epsilon$. |
A sequence $s$ is Cauchy if and only if for every $\epsilon > 0$, there exists $N$ such that for all $n \geq N$, we have $d(s_n, s_N) < \epsilon$. |
If a sequence $X$ satisfies the Cauchy condition, then it is Cauchy. |
If $X$ is a sequence in a metric space such that for every $\epsilon > 0$, there exists $M$ such that for all $m \geq M$ and all $n > m$, we have $d(X_m, X_n) < \epsilon$, then $X$ is a Cauchy sequence. |
If $X$ is a Cauchy sequence in a metric space, then for every $\epsilon > 0$, there exists an $M$ such that for all $m, n \geq M$, we have $d(X_m, X_n) < \epsilon$. |
A sequence in a metric space is Cauchy if and only if for every $j$, there exists an $M$ such that for all $m, n \geq M$, we have $d(x_m, x_n) < \frac{1}{j+1}$. |
A sequence $X$ is Cauchy if and only if for every $j \in \mathbb{N}$, there exists $M \in \mathbb{N}$ such that for all $m, n \geq M$, we have $|X_m - X_n| < \frac{1}{j+1}$. |
The sequence $1/n$ converges to $0$. |
A metric space is complete if and only if every Cauchy sequence converges to a point in the space. |
A metric space $X$ is totally bounded if and only if for every $\epsilon > 0$, there exists a finite set $K \subseteq X$ such that $X \subseteq \bigcup_{x \in K} B(x, \epsilon)$. |
A sequence converges if and only if it is Cauchy. |
If a Cauchy sequence has a convergent subsequence, then it converges. |
If $f$ is an increasing sequence of real numbers bounded above by $l$, and if for every $\epsilon > 0$, there exists an $n$ such that $l \leq f(n) + \epsilon$, then $f$ converges to $l$. |
If a sequence of real numbers is Cauchy, then it converges. |
If $f$ is a function from the reals to a first-countable topological space, and if for every sequence $X$ that converges to $\infty$, the sequence $f(X)$ converges to $y$, then $f$ converges to $y$ at $\infty$. |
If $f$ is a monotone function and $(f(n))_{n \in \mathbb{N}}$ converges to $y$, then $(f(x))_{x \in \mathbb{R}}$ converges to $y$ as $x \to \infty$. |
Suppose $f$ is a holomorphic function on an open set $s$ that does not contain a finite set of points $pts$. If $a$ and $b$ are two points in $s$ that are not in $pts$, then there exists a path $g$ from $a$ to $b$ that does not intersect $pts$ and such that $f$ is integrable along $g$. |
If $f$ is holomorphic on an open set $s$ and $g$ is a closed path in $s$ that does not pass through any of the points in a finite set $pts \subseteq s$, then the contour integral of $f$ along $g$ is equal to the sum of the winding numbers of $g$ around each point in $pts$ times the contour integral of $f$ around a circle around each point in $pts$. |
If $f$ is holomorphic on a connected open set $s$ and $g$ is a closed path in $s$ that does not pass through any of the points in a finite set $pts$, then the contour integral of $f$ along $g$ is equal to the sum of the contour integrals of $f$ along the circles centered at the points in $pts$ and with radii small enough to avoid the other points in $pts$. |
If $f$ is holomorphic on an open connected set $S$ and $g$ is a closed path in $S$ that does not pass through any of the poles of $f$, then the contour integral of $f$ along $g$ is equal to $2\pi i$ times the sum of the residues of $f$ at the poles of $f$ inside $g$. |
The argument principle states that the number of zeros of a holomorphic function $f$ inside a closed curve $C$ is equal to the number of poles of $f$ inside $C$ plus the winding number of $C$ around the origin. |
If $f$ is a holomorphic function on a connected open set $A$ that contains the closed disk of radius $r$ centered at the origin, and $f$ has no zeros in the open disk of radius $r$ centered at the origin, then the $n$th derivative of $f$ at the origin is equal to the sum of the residues of $f$ at the zeros of $f$ in the closed disk of radius $r$ centered at the origin, plus the integral of $f$ over the circle of radius $r$ centered at the origin. |
Suppose $f$ is a holomorphic function on a connected open set $A$ containing the closed ball of radius $r$ centered at the origin, and $S$ is a finite set of points in the open ball of radius $r$ centered at the origin. Suppose that $0$ is not in $S$. Then the sequence of coefficients of the Taylor series of $f$ centered at $0$ is asymptotically equal to the sequence of residues of $f$ at the points in $S$. |
Suppose $f$ is a holomorphic function on a connected open set $A$ that contains the closed ball of radius $r$ centered at the origin. Suppose $f$ has a power series expansion about the origin. Suppose $S$ is a finite set of points in the open ball of radius $r$ centered at the origin, and $0 \notin S$. Then the coefficients of the power series expansion of $f$ are asymptotically equal to the sum of the residues of $f$ at the points in $S$. |
If $f$ and $g$ are holomorphic functions on an open connected set $S$, and if $f$ has no zeros in $S$, then the number of zeros of $f + g$ in $S$ is equal to the number of zeros of $f$ in $S$. |
The Moebius function $f(z) = \frac{z - w}{1 - \overline{w}z}$ maps $0$ to $w$. |
The Möbius function is zero if the two arguments are equal. |
The Moebius function of zero is $-e^{it}w$. |
If $|w| < 1$ and $|z| < 1$, then $|\mu_t(w,z)| < 1$. |
If $|w| < 1$, then the Moebius function $f(z) = \frac{z-t}{1-\overline{t}z}$ is holomorphic on the unit disk. |
If $w_1$ and $w_2$ are complex numbers such that $w_1 = -w_2$ and $|w_1| < 1$, then the Moebius transformation $M_{w_1}$ composed with $M_{w_2}$ is the identity map. |
For any point $a$ in the unit disk, there exists a biholomorphism $f$ of the unit disk onto itself such that $f(a) = 0$. |
If $S$ is an open connected set containing $0$ and contained in the unit disc, and if for every holomorphic function $f$ on $S$ that is not identically zero, there exists a holomorphic function $g$ on $S$ such that $f(z) = g(z)^2$ for all $z \in S$, then there exist holomorphic functions $f$ and $g$ on $S$ and on the unit disc, respectively, such that $f(z) \in \mathbb{D}$ and $g(f(z)) = z$ for all $z \in S$, and $g(z) \in S$ and $f(g(z)) = z$ for all $z \in \mathbb{D}$. |
If $S$ is a simply connected set, then $S$ is connected and for any closed path $\gamma$ in $S$ and any point $z$ not in $S$, the winding number of $\gamma$ about $z$ is zero. |
If $f$ is holomorphic on a region $S$ and $g$ is a closed path in $S$, then $\int_g f(z) dz = 0$. |
If $f$ is a holomorphic function on a connected set $S$ and if $\gamma$ is a closed path in $S$, then $\int_\gamma f(z) dz = 0$. Then there exists a holomorphic function $h$ on $S$ such that $f = h'$. |
If $f$ is a holomorphic function on a connected set $S$ such that $f(z) \neq 0$ for all $z \in S$, then there exists a holomorphic function $g$ on $S$ such that $f(z) = e^{g(z)}$ for all $z \in S$. |
If $f$ is a holomorphic function on a set $S$ such that $f(z) \neq 0$ for all $z \in S$, then there exists a holomorphic function $g$ on $S$ such that $f(z) = g(z)^2$ for all $z \in S$. |
If $S$ is a connected non-empty proper subset of the complex plane, and if for every holomorphic function $f$ on $S$ with $f(z) \neq 0$ for all $z \in S$, there exists a holomorphic function $g$ on $S$ such that $f(z) = (g(z))^2$ for all $z \in S$, then there exist holomorphic functions $f$ and $g$ on $S$ and on the unit disc, respectively, such that $f(z) \in \mathbb{D}$ and $g(f(z)) = z$ for all $z \in S$, and $g(z) \in S$ and $f(g(z)) = z$ for all $z \in \mathbb{D}$. |
If $S$ is a nonempty set and either $S = \mathbb{C}$ or there exist holomorphic functions $f$ and $g$ such that $f$ maps $S$ into the unit disc, $g$ maps the unit disc into $S$, and $g \circ f$ and $f \circ g$ are the identity maps on $S$ and the unit disc, respectively, then $S$ is homeomorphic to the unit disc. |
If $S$ is empty or homeomorphic to the unit disc, then $S$ is simply connected. |
A simply connected open set is a connected open set with the following properties: The winding number of any closed path in the set is zero. The contour integral of any holomorphic function over any closed path in the set is zero. Any holomorphic function on the set has a primitive. Any holomorphic function on the set that is nonzero on the set has a logarithm. Any holomorphic function on the set that is nonzero on the set has a square root. The set is biholomorphic to the open unit disc. The set is homeomorphic to the open unit disc. |
If $S$ is an open subset of $\mathbb{C}$, then $S$ is contractible if and only if $S$ is simply connected. |
If $S$ is simply connected, then either $S$ is bounded and the frontier of $S$ is connected, or $S$ is unbounded and each component of the frontier of $S$ is unbounded. |
If $S$ is a connected set and $C$ is a component of the complement of $S$, then $C$ is unbounded. |
If $S$ is a connected set and every component of the complement of $S$ is unbounded, then the inside of $S$ is empty. |
If a connected set $S$ has no interior points, then $S$ is simply connected. |
A set $S$ is simply connected if and only if it is connected and either (1) $S$ is bounded and the frontier of $S$ is connected, or (2) $S$ is unbounded and every component of the complement of $S$ is unbounded. |
A bounded open set $S$ is simply connected if and only if $S$ and $-S$ are connected. |
If $f$ is a continuous function defined on a simply connected set $S$ and $f(z) \neq 0$ for all $z \in S$, then there exists a continuous function $g$ such that $f(z) = e^{g(z)}$ for all $z \in S$. |
If $f$ is a continuous function on a set $S$ such that $f(z) \neq 0$ for all $z \in S$, then there exists a continuous function $g$ on $S$ such that $f(z) = g(z)^2$ for all $z \in S$. |
If $S$ is connected and every continuous function $f$ from $S$ to $\mathbb{C}$ with $f(z) \neq 0$ for all $z \in S$ has a continuous square root, then $S$ is simply connected. |
A set $S$ is simply connected if and only if it is connected and for every continuous function $f$ defined on $S$ with $f(z) \neq 0$ for all $z \in S$, there exists a continuous function $g$ such that $f(z) = e^{g(z)}$ for all $z \in S$. |
If $S$ is locally connected or compact, then $S$ is Borsukian if and only if each component of $S$ is Borsukian. |
If each component of a set $S$ is Borsukian, then $S$ is Borsukian. |
A set $S$ is simply connected if and only if it is open, connected, and Borsukian. |
If $S$ is an open subset of the complex plane, then $S$ is Borsukian if and only if every component of $S$ is simply connected. |
If $S$ is an open or closed bounded set, then $S$ is Borsukian if and only if $-S$ is connected. |
A simply connected open set is either empty, the whole plane, or biholomorphic to the unit disk. |
If $p$ is a simple path, then the inside of the path image of $p$ is simply connected. |
If $S$ and $T$ are open, simply connected sets in the complex plane, and $S \cap T$ is connected, then $S \cap T$ is simply connected. |
If $p$ is a path and $\zeta$ is not in the image of $p$, then there exists a path $q$ such that $p(t) = \zeta + e^{q(t)}$ for all $t \in [0,1]$. |
If $p$ is a path and $\zeta$ is not in the image of $p$, then the winding number of $p$ around $\zeta$ is zero if and only if $p$ is homotopic to a constant path. |
If $p$ is a path and $\zeta$ is not in the image of $p$, then the winding number of $p$ around $\zeta$ is zero if and only if $p$ is homotopic to the constant path. |
If $p$ is a path and $\zeta$ is not in the image of $p$, then the winding number of $p$ around $\zeta$ is zero if and only if $p$ is homotopic to a constant path. |
If two paths start and end at the same point, then they have the same winding number around a point if and only if they are homotopic. |
If $p$ and $q$ are loops in the plane that do not pass through the point $\zeta$, then $p$ and $q$ are homotopic if and only if they have the same winding number around $\zeta$. |
If $x$ is an element of $M$, then $x$ is a subset of $\Omega$. |
Subsets and Splits