Function space

Example

Let $F$ be a field, let $X$ be a set, and let $V$ be a vector space over $F$. The set $V^X$ of all functions $f \colon X \to V$ is a vector space over $F$ with the pointwise operations $$(f + g)(x) = f(x) + g(x), \qquad (af)(x) = a\, f(x)$$ for $f, g \in V^X$, $a \in F$ and $x \in X$. The zero vector is the function with value $0$ at each point.