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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

On Burkholder's biconvex-function characterization of Hilbert spaces


Author: Jinsik Mok Lee
Journal: Proc. Amer. Math. Soc. 118 (1993), 555-559
MSC: Primary 46C15; Secondary 46B20, 46E40
MathSciNet review: 1159174
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Abstract: Suppose that $ {\mathbf{X}}$ is a real or complex Banach space with norm $ \vert \cdot \vert$. Then $ {\mathbf{X}}$ is a Hilbert space if and only if

$\displaystyle E\vert x + Y\vert \geqslant 1$

for all $ x \in {\mathbf{X}}$ and all $ {\mathbf{X}}$-valued Bochner integrable functions $ Y$ on the Lebesgue unit interval satisfying $ EY = 0$ and $ \vert Y\vert \geqslant 1$ a.e. This leads to a simple proof of the biconvex-function characterization due to Burkholder.

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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1993-1159174-6
PII: S 0002-9939(1993)1159174-6
Keywords: Biconvexity, $ \zeta $-convexity, Bochner integrable functions
Article copyright: © Copyright 1993 American Mathematical Society