On Von Neumann–jordan Constants

Abstract

In this note, we provide an example of a Banach space X for whichC N J (X ) = 1 that is not isomorphic to any Hilbert space, whereC N J (X ) denotes the infimum of all von Neumann-Jordan constants for equivalent norms of X .2000 Mathematics subject classification: primary 46B03; secondary 46B20.

An approximate version of the Jordan von Neumann theorem for finite-dimensional real normed spaces

“…For example, recent work by F. Wang has provided the final push to show that the von Neumann-Jordan constant C N J (X) satisfies C N J (X) ≤ sup{min{||x + y||, ||x − y||} : ||x|| = ||y|| = 1}, the latter expression being known as the James constant (see [2]). Slightly closer in flavor to the results presented here, a result of K. Hashimoto and G. Nakamura in [3] establishes that a different type of approximate Jordan von Neumann Theorem fails. They demonstrate that the modified von Neumann-Jordan constant C N J (X) = inf{C N J (X, | · |) : | · | is an equivalent norm to || · || on X} of value 1 is not sufficient to determine if a Banach space X is isomorphic with a Hilbert space.…”

“…the latter expression being known as the James constant (see [2]). Slightly closer in flavor to the results presented here, a result of K. Hashimoto and G. Nakamura in [3] establishes that a different type of approximate Jordan von Neumann Theorem fails. They demonstrate that the modified von Neumann-Jordan constant…”

Normed spaces equivalent to inner product spaces and stability of functional equations

“…There exist normed spaces with von Neumann-Jordan constants arbitrarily close to 1 which are not equivalent to any inner product space. Hashimoto and Nakamura [13] constructed a Banach space X such that C NJ (X) = 1 and X is not equivalent to any Hilbert space. The construction is as follows.…”