# 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 $\left({V^X, +, \circ}\right)_K$ is a vector space over $K$.