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