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.

Sources

• Seth Warner: Modern Algebra (1965)... (previous)... (next): $\S 26$: Example $26.4$