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. #m/def/anal/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 $| 𝑇 𝑓 | \leq (𝑏 - 𝑎) ‖ 𝑓 ‖ \infty$

#state/tidy | #lang/en | #SemBr