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

作者：Passer Benjamin^*

来源：Linear and Multilinear Algebra, 2015, 63(1): 68-77.

DOI：10.1080/03081087.2013.844234

• von Neumann-Jordan constant
• Banach-Mazur distance
• paralellogram law
• inner product
• 15A60
• 15A04
• 15A63

It is known that any normed vector space which satisfies the parallelogram law is actually an inner product space. For finite-dimensional normed vector spaces over , we formulate an approximate version of this theorem: if a space approximately satisfies the parallelogram law, then it has a near isometry with Euclidean space. In other words, a small von Neumann Jordan constant for yields a small Banach-Mazur distance with , . Finally, we examine how this estimate worsens as increases, with the conclusion that grows quadratically with .

• 出版日期2015-1-2