Functional analysis MOC

Bounded linear operator

A bounded linear operator is a linear map between TVSs that maps bounded subsets of to bounded subsets of , #m/def/anal/fun

Normed vector space

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

• is bounded
• is Lipschitz continuous at .
• is continuous anywhere.
• there exists such that for all (see Uniform continuity)

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

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