New characterizations of inner product spaces. (English) Zbl 0544.46016

Rassias, Themistocles M.

Bull. Sci. Math., II. Sér. 108, 95-99 (1984).

The purpose of this paper is to formulate and prove the following new characterizations of real inner product spaces. Necessary and sufficient conditions that the norm defined over a real vector space V is induced by an inner product are that:
(I) \(\| v+w\|^ n+\| v-w\|^ n\leq 2^{n-1}(\| v\|^ n+\| w\|^ n)\) for \(n>2\), or for any fixed \(n\geq 2,\)
(II) \(\sum^{n}_{i=1}\| v_ i-\frac{1}{n}\sum^{n}_{i=1}v_ i\|^ 2=\sum^{n}_{i=1}\| v_ i\|^ 2-n\| \frac{1}{n}\sum^{n}_{i=1}v_ i\|^ 2\) for \(v_ 1,v_ 2,...,v_ n\) vectors in a normed real vector space V.
These conditions as well as other results of the author concerning linear product spaces and the geometry of infinite dimensional spaces have been of importance for the solution of the longstanding problem of applying Morse theory on Hilbert manifolds to the Plateau’s problem [cf. the author (ed.), Global Analysis - Analysis on Manifolds, Teubner-Texte Math. 57 (1983; Zbl 0508.00007)].

MSC:

46C05	Hilbert and pre-Hilbert spaces: geometry and topology (including spaces with semidefinite inner product)
46B03	Isomorphic theory (including renorming) of Banach spaces
58E05	Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces

Keywords:

minimal area; elliptic norm; characterizations of real inner product spaces; applying Morse theory on Hilbert manifolds to the Plateau’s problem

Citations:

Zbl 0508.00007