Questions about spaces of functions, such as continuous functions between topological spaces or certain reproducing kernel Hilbert spaces. Does not concern equivalent classes of functions such as $L^p$ spaces.
A function space is a set of functions between two fixed sets. Often, the domain and/or codomain will have additional structure which is inherited by the function space. For example, the set of functions from any set $X$ into a vector space has a natural vector space structure given by pointwise addition and scalar multiplication. In other scenarios, the function space might inherit a topological or metric structure, hence the name function space.
Let $V$ be a vector space over a field $F$ and let $X$ be any set. The functions $X \to V$ can be given the structure of a vector space over $F$ where the operations are defined pointwise, that is, for any $f, g : X \to V$, any $x$ in $X$, and any $c$ in $F$, define
$$ \begin{aligned}(f+g)(x)&=f(x)+g(x)\\\\(c\cdot f)(x)&=c\cdot f(x)\end{aligned} $$
When the domain $X$ has additional structure, one might consider instead the subset (or subspace) of all such functions which respect that structure. For example, if $X$ is also a vector space over $F$, the set of linear maps $X \to V$ form a vector space over F with pointwise operations (often denoted $\operatorname {Hom}(X,V)$). One such space is the dual space of $V$: the set of linear functionals $V \to F$ with addition and scalar multiplication defined pointwise.
