# Function Space vs. Concrete Function

I need confirmation for my suspicion that

Space = A $\rightarrow$ B

defines the domain of all functions mapping from A to B while

f : A $\rightarrow$ B

defines one function out of that space.

Also, is the usage of the term "define" correct in the latter case? After all I'm not giving a concrete definition of $f$.

"Space = $A \to B$" is not a commonly accepted notation for spaces.

Rather, one writes $B^A$: we "make $A$ choices from $B$".

When vertical space is scarce (particularly with iterations) I have occasionally resorted to $[A \to B]$.

However, this is likely idiosyncratic and should be defined very clearly whenever it is used.

As to "$f: A \to B$", it names a (up until now arbitrary) function from $A$ to $B$. Because we haven't specified what $f$ is, we can only reason with what follows directly from $f \in B^A$ (e.g. that its domain is $A$, to name something trivial).

Given two nonempty sets $A$ and $B$ one consider maps $f:\>A\to B$. The last formula does not "define" $f$; it just tells the reader that $f$ is a mathematical object having the properties associated with a map from $A$ to $B$. E.g., you may consider $f$ as a subset of $A\times B$ having certain properties.

Now the set of all maps $f:\>A\to B$ is just a set, not a "space". It is quite common to denote this set by $B^A$. If $A$ and $B$ are finite then the set of maps $f:\>A\to B$ has in fact $|B|^{|A|}$ elements. But this notation has typographical disadvantages, and, above all, is very counterintuitive, since the maps in question go in the direction $A\to B$.

If you have to handle a lot of such sets of maps in your paper you could write ${\rm Map}(A,B)$ instead of $B^A$, in the same way as one denotes the set of all homomorphisms between two groups $G$ and $H$ by ${\rm Hom}(G,H)$.