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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Equations which characterize inner product spaces

Author: David A. Senechalle
Journal: Proc. Amer. Math. Soc. 40 (1973), 209-214
MSC: Primary 46C10
MathSciNet review: 0318851
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Abstract: It is shown that if $ N$ is a normed linear space and there is a point $ y$ of norm 1 such that an inequality of the type $ {a^2}\vert\vert x\vert{\vert^2} \leqq {\lim _{u \to 0}}G(\{ \vert\vert{b_i}ux + {c_i}y\vert\vert\} _{i = 1}^n) \leqq {b^2}\vert\vert x\vert{\vert^2}$ holds for all $ x$ in $ N$ (where $ 0 < a \leqq b$, the $ {c_i}$'s are nonzero and $ G$ and $ \vert\vert \cdot \vert\vert$ satisfy a certain twice-differentiability condition), then $ N$ is isomorphic to an inner product space and $ \inf \vert\vert T\vert\vert \cdot \vert\vert{T^{ - 1}}\vert\vert \leqq b/a$, where the infimum is taken over all linear homeomorphisms $ T$ between $ N$ and an inner product space. In the event that $ a = b = 1$, the inequality reduces to an equation which characterizes inner product spaces. An example shows that these results do not follow without the twice-differentiability condition on $ G$.

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Article copyright: © Copyright 1973 American Mathematical Society

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