How to read/interpret $\hat{f}(y)(x)$

Question

I'm currently in a point-set topology class and was presented with the following definition on a homework assignment.

If $f: X\times Y\to Z$ is a continuous function, we can define $\hat{f}: Y\to \mathcal{C}(X,Z)$ by $\hat{f}(y)(x)=f(x,y)$.

I understand that $\hat{f}$ is a function that takes points in $Y$ and maps them to continuous functions from $X$ to $Z$, but I am unsure how I should read this notation.

If $y$ is the input for $\hat{f}$, what is the role of $x$ in $\hat{f}(y)(x)$? Is it simply that we choose an arbitrary $x$ to "attach to" $\hat{f}$ so that we can have the equality $\hat{f}(y)(x)=f(x,y)$ for some preexisting $f$?

Answer 1

$\hat{f}$ is defining a mapping from $Y$ into the space of continuous functions ${\cal C}(X,Z)$. If you picked an element from that space, you'd probably name that function $\varphi$. So $\hat{f}(y) = \varphi$. Now $\varphi:X\rightarrow Z$ so $\varphi$ operates on some $x\in X$ and yields a point in $Z$: thus $\varphi(x)=z$ for $z\in Z$. However, there's no real need to have $\varphi$ in there in these definitions, it's syntactic sugar$^*$. So we replace $\varphi$ by what it is and we get $[\hat{f}(y)](x)$ Then, because it's quicker and it's not really ambiguous, we drop the square brackets to get the notation you're asking about.

When you get to $f(x,y) = \hat{f}(y)(x)$ you're now free to choose $x\in X$ and $y \in Y$ as you want; neither needs to be fixed. The original formulation fixed $y$ (in order to choose $\varphi = \hat{f}(y)$) and then varied $x$, but for any given $y$ there's a whole space ($X$) of $x$'s to choose from now.

Reading it is a little tricky: I would suggest "the (continuous) function $\hat{f}$ of $y$ operating on a variable $x$ is given by a function $f$ of two variables $x$ and $y$" if you're reading it aloud or in your head.

$^*$ When you're first encountering this notation, $\hat{f}(y)=\varphi$ seems more familiar, but once you've had practice $\hat{f}(y)(x)$ is a good reminder that you're working with function spaces. You'll see this kind of notation used a lot in Functional Analysis and anything to do with Banach spaces.

Answer 2

Given $y \in Y$, $\hat{f}(y)$ is to be a function $\hat{f}(y): X \to Z$. Therefore, to define it, you should make sense of it on each point $x \in X$. So we write $\hat{f}(y)(x)=f(x,y)$ to do so.

In summary, the $y$ is fixed first, then you must consider what it does for each $x$ in order to well-define the function. This process may be done for all $y$, so this describes $\hat{f}: Y \to \mathcal{C}(X,Z)$.