Confusion between {..|..} and {...such that..} I came across the following equation in Mathematics for Machine Learning by Garret Thomas [page 9]
$$ range(T) = \{w \in W |\ \exists v \in V such \ that \ Tv=w \}$$
I was under the impression that $\{..|..\}$ meant such that. Could some please explain to me what exactly it means in this context?
 A: I read it as 'where' or 'for which'. So I would read you described set as "all w in W, for which, there exists an v in V such that Tv=w".
A: As you say, if you just write that set builder notation out in plain language, with two "such that"s, then it is ambiguous.
But you can remove the ambiguity by putting parentheses into your plain language, like this:

The set of all $w$ in $W$ such that (there exists $v$ in $V$ such that $Tv=w$).

A: Say your function is $T\colon V\to W$. Then the codomain of this function is $W$ and the range of $T$ is the subset of all $w\in W$ which are actually realized by the function $T$. 
That is, $w\in\textrm{range}(T)$ if and only if there exists an element $v\in V$ with $T(v)=w$. Note that $W=\textrm{range}(T)$ is not necessarily true, it might be that $\textrm{range}(T)\subsetneq W$. Therefore the elements in $\textrm{range}(T)$ need to be characterized by a special property. 
So in the notation $\left\{\cdot | \cdot\right\}$ or $\left\{\cdot : \cdot\right\}$, the part after the "$|$" or the "$:$" simply describes the property that an element of $W$ needs to satisfy in order to be contained in this set. Whether you read this as such that or for which or with the property that is your personal decision and a matter of taste.
