Vol. 74, No. 1, 1978

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ISSN: 0030-8730
Banach spaces which satisfy linear identities

Bruce Reznick

Vol. 74 (1978), No. 1, 221–233
Abstract

In 1935, Jordan and von Neumann proved that any Banach space which satisfies the parallelogram law

∥x+ y∥2 + ∥x − y∥2 = 2(∥x∥2 + ∥y∥2)
for all elements x and y
(1)
must be a Hilbert space.

Subsequent authors have found norm conditions weaker than (1) which require a Banach space to be a Hilbert space. Notable examples include the results of Day, Lorch, Senechalle and Carlsson.

In this paper, we study nontrivial linear identities such as

∑m                         p
ak∥ck(0)x0 + ⋅⋅⋅+ ck(n)xn∥ = 0 for all elements xi
k=0
(2)

on a Banach space X.

Mathematical Subject Classification 2000
Primary: 46B99
Milestones
Received: 5 May 1977
Revised: 27 July 1977
Published: 1 January 1978
Authors
Bruce Reznick
Department of Mathematics and Center for Advanced Study
University of Illinois at Urbana-Champaign
1409 W. Green Street
327 Altgeld Hall
Urbana IL 61801-2975
United States