Measure, Integration & Real Analysis · Problem 9A.10

Linear Maps on Hilbert Spaces — Sheldon Axler

Mathematics — Measure Theory · total-variation-norm-not-inner-product

Problem

Suppose (X,S)(X, \mathcal{S}) is a measurable space with S≠{∅,X}\mathcal{S} \neq \{\emptyset, X\}. Prove that the total variation norm on MF(S)\mathcal{M}_{\mathbf{F}}(\mathcal{S}) does not come from an inner product. In other words, show that there does not exist an inner product ⟨⋅,⋅⟩\langle \cdot, \cdot \rangle on MF(S)\mathcal{M}_{\mathbf{F}}(\mathcal{S}) such that ∥ν∥=⟨ν,ν⟩1/2\|\nu\| = \langle \nu, \nu \rangle^{1/2} for all ν∈MF(S)\nu \in \mathcal{M}_{\mathbf{F}}(\mathcal{S}), where ∥⋅∥\|\cdot\| is the usual total variation norm on MF(S)\mathcal{M}_{\mathbf{F}}(\mathcal{S}).

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