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

 * Module of All Mappings

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