Functional analysis MOC

Bounded linear operator

A bounded linear operator is a linear map 𝑇 :𝑋 →𝑌 between TVSs that maps bounded subsets of 𝑋 to bounded subsets of 𝑌, fun

Normed vector space

Given two normed vector spaces over the same field 𝕂, a linear map 𝑇 :𝑋 →𝑌 is continuous with respect to the induced topology iff: norm

• 𝑇 is bounded
• 𝑇 is Lipschitz continuous at ⃗𝟎.
• 𝑇 is continuous anywhere.
• there exists 𝑘 ∈ℝ such that ‖𝑇⃗𝐯‖ ≤𝑘‖⃗𝐯‖ for all ⃗𝐯 ∈𝑋 (see Uniform continuity)

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

‖𝑇⃗𝐱‖≤‖𝑇‖op‖⃗𝐱‖

Proof

proof

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