The Continuous Functions with Compact Support Form a Real Normed Space under the Uniform Norm

In the setting of Locally Compact Metric Spaces and Compactly Supported Functions: Standing Notation, let $(X,d)$ be a metric space, and let $C_{c}(X)$ carry the sum, the scalar multiplication and the uniform norm $\|\cdot\|_{u}$ of The Space of Continuous Functions with Compact Support and the Uniform Norm (ii) and The Space of Continuous Functions with Compact Support and the Uniform Norm (iii).

(i) Vector space. $C_{c}(X)$ is a real vector space; its zero vector is the constant function $0$, and the negative of $f\in C_{c}(X)$ is $(-1)f$.

(ii) Normed space. $(C_{c}(X),\|\cdot\|_{u})$ is a real normed space.

(iii) Bounds. For every $f\in C_{c}(X)$, $|f(x)|\le\|f\|_{u}$ for every $x\in X$; and if $C\ge0$ is a real number with $|f(x)|\le C$ for every $x\in X$, then $\|f\|_{u}\le C$.

(iv) Uniform convergence. Let $(f_{m})$ be a sequence in $C_{c}(X)$ and let $f\in C_{c}(X)$. Then

$$\|f_{m}-f\|_{u}\to0$$

if and only if $(f_{m})$ converges uniformly to $f$ on $X$.