Functions to a Field form a Vector Space

Last updated Nov 1, 2022

Let $S$ be a Set and $F$ some Field. Then $\mathcal{F} = {f : S \to F}$ is a Vector Space with

• $(f +_\mathcal{F} g)(x) = f(x) +_F g(x)$ for $x \in S$, $f,g \in \mathcal{F}$.
• $(c *\mathcal{F} f)(x) = c *{F} f(x)$ for $c \in F$, $f \in \mathcal{F}$.

Combine Fields are Vector Spaces and Functions to a Vector Space form a Vector Space. $\blacksquare$