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.

Remarks

For $X = \{1, \dots, n\}$ and $V = F$, this is the coordinate space $F^n$. The continuous functions $[0, 1] \to \R$ form a subspace of $\R^{[0, 1]}$. If $X$ is infinite and $V \ne 0$, then $V^X$ is infinite-dimensional.

Explanations

L1 human

We add two functions by adding their values at each point. We scale a function by scaling its values. With these rules, functions behave like vectors, and the zero function is the zero vector.