Space bounded Turing Machine - clarification on Computational Complexity (book: Arora-Barak ) question 4.1
I do not understand what you are trying to say after "--", but here is what you need to do:

Hopefully you understand the idea of turing machine encoding: i.e. you can assign a unique identifier (i.e. "label") $\alpha$ to each turing machine. So by $M_\alpha$ we mean the Turing machine labeled by $\alpha$. So you need to design a single turing machine $SU$ which takes three inputs $\alpha$, $t$ and $x$ ($\alpha$ is a turing machine label, $t$ is a space bound, and $x$ is an input string) and:


if $M_\alpha$ uses more than $t$ bits of workspace on input $x$ before it halts, then we have no requirement for $SU$
if $M_\alpha$ uses at most $t$ bits of workspace on input $x$ we have two requirements for $SU$:


$SU$ uses at most $Ct$ units of workspace where $C$ is a function of $\alpha$ but not $t$ or $x$
$SU$ outputs $M_\alpha(x)$



The idea is that for any machine and any input, $SU$ can simulate that machine while using only slightly more space than the original machine. Note that $t$ is a bound on the space that $M_\alpha$ uses, not the space that $SU$ uses. $SU$ cannot be exactly as space efficient as the original machine because simulation requires some overhead, but the point is that the overhead is small as $C$ is fixed for every $\alpha$, no matter how large $x$ is in size.

Theorem 1.9. also has some overhead: a machine which stops in $T$ time steps is simulated in $CT\log T$ time steps where $C$ is again some constant independent of $T$ or of the input string.