Partial Mielnik spaces and characterization of uniformly convex spaces

Blass, A. R.; Stanojevic, C. V.

doi:10.1090/S0002-9939-1976-0394121-6

Partial Mielnik spaces and characterization of uniformly convex spaces
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by A. R. Blass and C. V. Stanojevic PDF

Proc. Amer. Math. Soc. 55 (1976), 75-82 Request permission

Abstract:

We characterize uniform convexity of normed linear spaces in terms of a functional inequality generalizing Clarkson’s inequality for ${L_p}$ spaces. This inequality can be interpreted as saying that the unit sphere of the space carries a structure slightly weaker than a probability space in the sense of Mielnik. From this point of view, our result is analogous to an earlier characterization of inner product spaces. We also investigate briefly the abstract concept of partial probability space suggested by the main result.

References

Garrett Birkhoff and John von Neumann, The logic of quantum mechanics, Ann. of Math. (2) 37 (1936), no. 4, 823–843. MR 1503312, DOI 10.2307/1968621
James A. Clarkson, Uniformly convex spaces, Trans. Amer. Math. Soc. 40 (1936), no. 3, 396–414. MR 1501880, DOI 10.1090/S0002-9947-1936-1501880-4

Topologische Lineare Räume

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Bogdan Mielnik, Geometry of quantum states, Comm. Math. Phys. 9 (1968), 55–80. MR 231603
Č. V. Stanojević, Mielnik’s probability manifolds and inner product spaces, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 18 (1970), 531–532 (English, with Russian summary). MR 273380
C. V. Stanojevic, Mielnik’s probability spaces and characterization of inner product spaces, Trans. Amer. Math. Soc. 183 (1973), 441–448. MR 328562, DOI 10.1090/S0002-9947-1973-0328562-1

Additional Information

Journal: Proc. Amer. Math. Soc. 55 (1976), 75-82
DOI: https://doi.org/10.1090/S0002-9939-1976-0394121-6
MathSciNet review: 0394121