From: Stephen Montgomery-Smith
Subject: Re: Q: "Hilbertizable" spaces
Date: Tue, 09 Feb 1999 10:14:02 -0600
Newsgroups: sci.math
To: "Volker W. Elling"
Keywords: Kwapien's theorem: norms equivalent to Hilbert space
Volker W. Elling wrote:
>
> Hello,
>
> there is a theorem stating that a norm is generated by a scalar product
> if
> and only if it satisfies the parallelogram inequality. However, in
> approximation theory and other areas, people care more about
> _equivalent_
> norms (i.e. |x|_1 \leq c|x|_2 \leq C|x|_1 means ||_1 and ||_2 are
> equivalent)
> than equal norms. There are examples for Banach spaces whose norms are
> equivalent but not equal to Hilbert space norms, for example
> (IR^2, ||_1)
> where
> |x|_1 := |x_1| + |x_2|
> Since IR^2 is finite-dimensional, ||_1 is equivalent to the euclidean
> norm;
> obviously they are not equal.
>
> Does anybody know anything about characterizations of norms that are
> equivalent to scalar products ?
>
> -- Volker Elling
> -- Informatik/Mathematik, RWTH Aachen -
> http://lem.stud.fh-heilbronn.de/~elling
Hi, there is a famous result due to Kwapien that states that a norm
on a Banach space is equivalent to a Hilbert space if and only
if it satisfies an approximate n-fold parallelogram law, that is,
if there is a constant c such that for all vectors x_1, x_1, ..., x_n
one has that
c^{-1} sum_{i=1}^n ||x_i||^2
<=
Average || sum_{i=1}^n r_i x_i ||^2
<=
c sum_{i=1}^n ||x_i||^2
where the average is taken over all signs r_i = plus/minus 1.
This is considered one of the central results of Banach space theory.
You can find the statement and proof of this result in a number of books,
including
Gilles Pisier, Factorization of Linear Operators and Geometry of Banach
Spaces, Regional Conference Series in Math 60, A.M.S. 1986.
Or Kwapien's original paper: Studia Math 44 (1972) 583-595.
This result is usually stated thus: A Banach space is isomorphic
to a Hilbert space if and only if it is of type 2 and cotype 2.
--
Stephen Montgomery-Smith stephen@math.missouri.edu
307 Math Science Building stephen@showme.missouri.edu
Department of Mathematics stephen@missouri.edu
University of Missouri-Columbia
Columbia, MO 65211
USA
Phone (573) 882 4540
Fax (573) 882 1869
http://math.missouri.edu/~stephen