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For any nonzero z0, there is a bounded functional f with ||f||=1 and f(z0)=||z0||.
Let be a normed space and let .
Corollary: There exists a bounded linear functional such that
This is a direct consequence of separation of a point and a subspace by applying it to and .
Let be a vector space over (or ). A norm on is a function such that for all and scalars :
A pair is called a normed vector space.
Let be a normed space over or . A linear map (where ) is a linear functional.
The functional is bounded if there exists such that
The norm (operator norm) of is
Let be a normed space, let be a subspace, and let satisfy
Theorem (separating a point and a subspace): There exists a bounded linear functional such that
Context: This is a geometric consequence of Hahn–Banach in normed spaces. It produces a continuous hyperplane through that separates from .
Proof sketch (idea): Define a linear functional on (see direct sum) by mapping and bound its norm using the distance assumption; then extend it by Hahn–Banach.