Bounded linear operator

A bounded linear operator is a linear map $𝑇 : 𝑋 \to 𝑌$ between TVSs that maps bounded subsets of $𝑋$ to bounded subsets of $𝑌$ , #m/def/anal/fun

Normed vector space

Given two normed vector spaces over the same field $𝕂$ , a linear map $𝑇 : 𝑋 \to 𝑌$ is continuous with respect to the induced topology iff: #m/thm/norm

$𝑇$ is bounded
$𝑇$ is Lipschitz continuous at $⃗ 𝟎$ .
$𝑇$ is continuous anywhere.
there exists $𝑘 \in ℝ$ such that $‖ 𝑇 ⃗ 𝐯 ‖ \leq 𝑘 ‖ ⃗ 𝐯 ‖$ for all $⃗ 𝐯 \in 𝑋$ (see Uniform continuity)

where the lowest possible $𝑘$ as its Operator norm. Therefore we may rephrase the second condition as

‖ 𝑇 ⃗ 𝐱 ‖ \leq ‖ 𝑇 ‖ o p ‖ ⃗ 𝐱 ‖

Proof

#missing/proof

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