Definition:Vector Space of All Mappings

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Definition

Let $\struct {K, +, \circ}$ be a division ring.

Let $\struct {G, +_G, \circ}_K$ be a $K$-vector space.

Let $S$ be a set.

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

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

$+_G'$ is the operation induced on $G^S$ by $+_G$
$\forall \lambda \in K: \forall f \in G^S: \forall x \in S: \map {\paren {\lambda \circ f} } x = \lambda \circ \paren {\map f x}$

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

Examples

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