Definition:Vector Space of All Mappings

Theorem
Let $\left({K, +, \circ}\right)$ be a division ring.

Let $\left({G, +_G, \circ}\right)_K$ be a $K$-vector space.

Let $S$ be a set.

Let $G^S$ be the set of all mappings from $S$ to $G$.

Then $\left({G^S, +_G', \circ}\right)_K$ is a $K$-vector space, where:


 * $+_G'$ is the operation induced on $G^S$ by $+_G$


 * $\forall \lambda \in K: \forall f \in G^S: \forall x \in S: \left({\lambda \circ f}\right) \left({x}\right) = \lambda \circ f \left({x}\right)$

This is the $K$-vector space $G^S$ of all mappings from $S$ to $G$.

The most important case of this example is when $\left({G^S, +_G', \circ}\right)_K$ is the $K$-vector space $\left({K^S, +_K', \circ}\right)_K$.

Also see

 * Definition:Module of All Mappings

Proof
Follows directly from Module of All Mappings is Module and the definition of vector space.