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Hilbert spaces have the Radon--Nikodym property
Example
Assume the Axiom of Choice. Every real or complex Hilbert space has the Radon--Nikodym property.
Facts & Assumptions
The Axiom of Choice holds and implies the Axiom of Countable Choice (The Axiom of Choice, AC supplies the countable and dependent choices used in Banach integration).
Under Countable Choice, every complete real or complex inner-product space is reflexive (Hilbert spaces are reflexive by Riesz representation).
Under AC, every real or complex reflexive Banach space has the Radon--Nikodym property (Reflexive spaces have the Radon--Nikodym property).
Verification
Given: AC and a real or complex Hilbert space .
Propagate the choice assumption to the reflexivity supplier. By [A1], the assumed AC supplies the Countable Choice required by [L1]. No separate choice assumption is introduced.
Pass from Hilbert structure to RNP. A Hilbert space is complete for its inner-product norm, so [L1] makes reflexive. It is therefore a reflexive Banach space, and [L2], under the same AC hypothesis, gives the Radon--Nikodym property.
Audit the scope and degenerate cases. [A1, L1, L2, step 1.1, step 2.1] The argument applies to both real and complex scalar fields and to Hilbert spaces of arbitrary dimension and separability. For the zero Hilbert space, reflexivity and RNP are included in the two suppliers and the density condition is vacuous at zero vector measures. Completeness is essential to the word ``Hilbert'' here; no assertion is made for an incomplete inner-product space. The only choice propagation is AC to Countable Choice in step 1.1 and AC into [L2].
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Sources
- Gilles Pisier, Martingales in Banach Spaces (standard reference, not scraped)
- Haim Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations (standard reference, not scraped)