DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-generatedverified 2026-09-24 (gpt-6-sol)

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Reflexivity is surjectivity of the canonical map

Definition

Let $K = R$ or $C$ . A Banach space $X$ is reflexive if its canonical map $J_{X} : X \to X^{* *}$ from The canonical evaluation map into the bidual is surjective. Assuming the Axiom of Choice (The Axiom of Choice), The canonical bidual map is an isometry shows that this map is already an isometric embedding. Surjectivity means that every bounded linear functional on $X^{*}$ is evaluation at a vector of $X$ . Merely specifying some isomorphism between $X$ and $X^{* *}$ is not this definition.

Depends on

Used by

Hilbert spaces are reflexive Corollary
Separable reflexive space has separable dual Corollary
c₀ is not reflexive Counterexample
The canonical bidual map of c0 misses the constant sequence Counterexample
Reflexivity of ℓᵖ and Lᵖ Example
A bounded sequence in a reflexive Banach space has a weakly convergent subsequence Lemma
James nonreflexivity sequence separated from an annihilator Lemma
The zero-boundary W1p space is separable and reflexive Lemma
W^1,p(Omega) is reflexive for 1<p<infinity Lemma
Weak compactness of bounded space-time Sobolev gradients Lemma
A Banach space is reflexive if and only if its dual is reflexive Theorem
Closed subspaces of reflexive spaces are reflexive Theorem
Conditional monotone finite-element convergence Theorem
Higher eigenvalues by orthogonality-constrained minimisation Theorem
Hilbert spaces are reflexive by Riesz representation Theorem
James reflexivity theorem Theorem
Milman–Pettis theorem Theorem
Quotients of reflexive spaces are reflexive Theorem
Reflexive approximation property implies metric approximation property Theorem
Reflexive iff unit ball weakly compact Theorem
Reflexive spaces have the Radon--Nikodym property Theorem
Reflexivity of Lp for one less p less infinity Theorem
The canonical image of James space has codimension one Theorem
The direct method for convex integral functionals Theorem
The direct method in a reflexive Banach space Theorem
The direct method on a weakly closed constraint set Theorem
The first Dirichlet eigenfunction by constrained minimisation Theorem

Dependency tree · two levels

11 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Bühler–Salamon, Functional Analysis, Definition 2.70, p.89 (standard reference, not scraped)