The Real Vector Space of Real-Valued Functions on a Set

In the setting of The Real Numbers: Standing Notation and Background, let $X$ be a set and let $\mathrm{Map}(X,\mathbb{R})$ denote the set of all maps from $X$ to $\mathbb{R}$, two such maps being equal exactly when they take the same value at every point of $X$. For $f,g\in\mathrm{Map}(X,\mathbb{R})$ and $\lambda\in\mathbb{R}$ define the pointwise sum $f+g$ and the pointwise scalar multiple $\lambda f$ to be the maps given by

and let $0_{X}$ denote the map taking the value $0$ at every point of $X$. Then the following hold.

1. (The space of all real-valued maps)¶ The set $\mathrm{Map}(X,\mathbb{R})$, equipped with the pointwise sum and the pointwise scalar multiple, is a vector space over $\mathbb{R}$. Its zero vector is $0_{X}$; the additive inverse $-f$ of $f$ is the map $x\mapsto-f(x)$; and the difference $f-g$ is the map $x\mapsto f(x)-g(x)$.

2. (Subspace criterion)¶ Let $W$ be a subset of $\mathrm{Map}(X,\mathbb{R})$ such that $0_{X}\in W$, and such that $f+g\in W$ and $\lambda f\in W$ for all $f,g\in W$ and every $\lambda\in\mathbb{R}$. Then $W$ is a linear subspace of $\mathrm{Map}(X,\mathbb{R})$, and $W$, equipped with the pointwise sum and the pointwise scalar multiple restricted to $W$, is itself a vector space over $\mathbb{R}$ with zero vector $0_{X}$.