Field/Set of mappings/Vector space/Example

Let ${}K$ denote a field and let ${}I$ denote a set. We consider the set of functions from ${}I$ to ${}K$, that is

${}V=\operatorname {Map} \,{\left(I,K\right)}={\left\{f\mid f:I\rightarrow K{\text{ mapping}}\right\}}\,.$

This set is with componentwise addition, where the sum of two functions ${}f$ and ${}g$ is given as

${}(f+g)(z):=f(z)+g(z)\,$

for ${}z\in I$, and with scalar multiplication defined by

${}(sf)(z):=sf(z)\,,$

a vector space.