infinity norm for continuous bounded function space

[[concept]]

Proposition

Then $C_{\infty} (X)$ is a vector space and we can define the norm

| | u | |_{\infty} = sup_{x \in X} | u (x) | $ $ o n $ C_{\infty} (X) $

Proof

NTS that this is indeed a norm by verifying each of the properties. identifiability and homogeneity are satisfied from the definition of the norm. It suffices then to show that the triangle inequality holds.

If $(u, v) \in C_{\infty} (X)$ then for all $x \in X$

\begin{aligned} | u (x) + v (x) | & \leq | u (x) | + | v (x) | by △ inequality \\ \leq | | u | |_{\infty} + | | v | |_{\infty} \\ ⟹ | | u + v | |_{\infty} & = sup_{x \in X} | u (x) + v (x) | \\ \leq | | u | |_{\infty} + | | v | |_{\infty} \end{aligned}

Note

Convergence in this norm means

\begin{aligned} u_{n} \to u in C_{\infty} (X) & ⟺ | | u_{n} - u | |_{\infty} \to 0 as n \to \infty \\ ⟺ \forall ϵ > 0, \exists N \in N s.t. \forall n \geq N, \forall x \in X, | u_{n} (x) - u (x) | < ϵ \\ ⟺ u_{n} \to u uniformly on X \end{aligned}

Thus convergence in this metric is uniform convergence when the functions $u$ are bounded and continuous.

Mentions

File	Last Modified
Functional Analysis Lecture 1	2025-09-02

Created 2025-05-27 ֍ Last Modified 2025-09-15