Measure, Integration & Real Analysis · Problem 6D.13

L^p Spaces — Sheldon Axler

Mathematics — Measure Theory · unique-extension-bounded-functional

Problem

Suppose VV is a normed vector space, UU is a subspace of VV, and ψ:U→R\psi: U \to \mathbf{R} is a bounded linear functional. Prove that ψ\psi has a unique extension to a bounded linear functional φ\varphi on VV with ∥φ∥=∥ψ∥\|\varphi\| = \|\psi\| if and only if sup⁡f∈U(−∥ψ∥∥f+h∥−ψ(f))=inf⁡g∈U(∥ψ∥∥g+h∥−ψ(g))\sup_{f \in U}\bigl(-\|\psi\| \|f + h\| - \psi(f)\bigr) = \inf_{g \in U}\bigl(\|\psi\| \|g + h\| - \psi(g)\bigr) for every h∈V∖Uh \in V \setminus U.

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