Operator norm

The operator norm is a norm on a Space of bounded linear operators. Informally, it is the maximum factor by which it lengthens vectors. vec

‖ 𝑇 ‖ o p = s u p {‖ 𝑇 𝑥 ‖ ‖ 𝑥 ‖ : 𝑥 \in 𝑋 ∖ {0}} = s u p {‖ 𝑇 𝑥 ‖ : ‖ 𝑥 ‖ = 1}

Example

Consider the linear map $𝑇 : C (𝑎, 𝑏) \to ℝ$ with $‖ \cdot ‖ \infty$ and $‖ \cdot ‖ 2$ respectively where
$𝑇 𝑓 = \int 𝑏 𝑎 𝑓 (𝑥) 𝑑 𝑥$
then $‖ 𝑇 ‖ o p = 𝑏 - 𝑎$ , since clearly $| 𝑇 𝑓 | \leq (𝑏 - 𝑎) ‖ 𝑓 ‖ \infty$

tidy | en | SemBr

Operator norm

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