Editing Function space (section)

==In linear algebra==
{{See also|Vector space#Function spaces}}
{{Unreferenced section|date=November 2017}}

Let {{var|F}} be a [[Field (mathematics)|field]] and let {{var|X}} be any set. The functions {{var|X}} → {{var|F}} can be given the structure of a vector space over {{var|F}} where the operations are defined pointwise, that is, for any {{var|f}}, {{var|g}} : {{var|X}} → {{var|F}}, any {{var|x}} in {{var|X}}, and any {{var|c}} in {{var|F}}, define
<math display="block">
\begin{align}
  (f+g)(x) &= f(x)+g(x) \\
  (c\cdot f)(x) &= c\cdot f(x)
\end{align}
</math>
When the domain {{var|X}} has additional structure, one might consider instead the [[subset]] (or [[Linear subspace|subspace]]) of all such functions which respect that structure. For example, if {{var|V}} and also {{var|X}} itself are vector spaces over {{var|F}}, the set of [[Linear map|linear maps]] {{var|X}} → {{var|V}} form a vector space over {{var|F}} with pointwise operations (often denoted [[Hom set|Hom]]({{var|X}},{{var|V}})). One such space is the [[dual space]] of {{var|X}}: the set of [[Linear form|linear functionals]] {{var|X}} → {{var|F}} with addition and scalar multiplication defined pointwise.

The cardinal [[dimension]] of a function space with no extra structure can be found by the [[Erdős–Kaplansky theorem]].