Normed spaces isometric to their duals

DefinitionMazur's Problem 91 (Scottish Book), Banach-space form#✓fully proved

This is the proposition that every finite-dimensional real normed space \(E\) that is linearly isometric to its dual space \(E^*\) (the continuous linear functionals on \(E\) with the operator norm) is an inner product space, that is, its norm is induced by some inner product on \(E\).

Mazur's original formulation asks whether, for every \(n\), a convex body in \(\mathbb{R}^n\) that is affinely equivalent to its conjugate (polar) body is necessarily an ellipsoid.

def ScottishBookProblem91Claim : Prop :=
  ∀ (n : ℕ) (E : Type*) [NormedAddCommGroup E] [NormedSpace ℝ E]
    [FiniteDimensional ℝ E],
    Module.finrank ℝ E = n →
    (∃ _ : E ≃ₗᵢ[ℝ] StrongDual ℝ E, True) →
    ∃ _ : InnerProductSpace ℝ E, True

Used by 1

Used byMazur's Problem 91 has a negative answer

TheoremMazur's Problem 91 has a negative answer#●ready to prove

There is a finite-dimensional real normed space \(E\) that is linearly isometric to its dual space \(E^*\) (the continuous linear functionals on \(E\) with the operator norm) but whose norm is not induced by any inner product on \(E\). Thus the Banach-space form of Mazur's Problem 91 is false.

theorem_wanted mazur91_refuted : ¬ ScottishBookProblem91Claim

Sources

• S. Mazur, Problem 91 of the Scottish Book. The typescript itself records the negative answer when n is even (for example, ℓ¹ 2, whose unit ball is a diamond, is linearly isometric to its dual ℓ^∞ 2 but is not a Hilbert space, since its norm does not satisfy the parallelogram law); the odd-dimensional case is not needed to refute the universal claim.

Uses 1

Statement usesMazur's Problem 91 (Scottish Book), Banach-space form

1 fully proved1 ready to prove

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