Mappings to Vector Space form Vector Space
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Theorem
Let $X$ be a non-empty set.
Let $V$ be a vector space over a field (or division ring) $K$.
Let $V^X$ denote the set of all mappings from $X$ to $V$.
Let $+$ denote pointwise addition on $V^X$.
Let $\circ$ denote pointwise ($K$)-scalar multiplication on $V^X$.
Then $\struct {V^X, +, \circ}_K$ is a vector space over $K$.
Proof
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