The Space of Continuous Functions with Compact Support and the Uniform Norm

In the setting of Locally Compact Metric Spaces and Compactly Supported Functions: Standing Notation, let $(X,d)$ be a metric space.

(i) The space $C_{c}(X)$. $C_{c}(X)$ is the set of continuous functions $f:X\to\mathbb{R}$ with compact support (a set, as a subclass of the set of functions $X\to\mathbb{R}$, by Subclasses of Sets Are Sets, the Union and Power Set of a Set Exist Uniquely, Binary Unions of Sets Are Sets, and the Universal Class Is Proper (i)).

(ii) Operations. The sum of $f,g\in C_{c}(X)$ is the pointwise sum $f+g$, and the scalar multiple of $f\in C_{c}(X)$ by $c\in\mathbb{R}$ is the pointwise multiple $cf$; both lie in $C_{c}(X)$ by Continuous Functions with Compact Support: the Zero Function, Closure under Operations, Vanishing off the Support, Existence of the Maximum Modulus, and Uniform Continuity (ii).

(iii) Uniform norm. The uniform norm of $f\in C_{c}(X)$ is the greatest element

$$\|f\|_{u}=\max\bigl(\{0\}\cup\{|f(x)|:x\in X\}\bigr),$$

which exists by Continuous Functions with Compact Support: the Zero Function, Closure under Operations, Vanishing off the Support, Existence of the Maximum Modulus, and Uniform Continuity (iv).