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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



$ G\sb{2n}$ spaces

Author: Donald O. Koehler
Journal: Trans. Amer. Math. Soc. 150 (1970), 507-518
MSC: Primary 46.15; Secondary 47.00
MathSciNet review: 0262806
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Abstract: A complex normed linear space $ X$ will be called a $ {G_{2n}}$ space if and only if there is a mapping $ \left\langle { \cdot , \ldots , \cdot } \right\rangle $ from $ {X^{2n}}$ into the complex numbers such that: $ {x_k} \to \left\langle {{x_1}, \ldots ,{x_{2n}}} \right\rangle $ is linear for $ k = 1, \ldots ,n;\left\langle {{x_1}, \ldots ,{x_{2n}}} \right\rangle = {\left\langle {{x_{2n}}, \ldots ,{x_1}} \right\rangle ^ - }$; and $ {\left\langle {x, \ldots ,x} \right\rangle ^{1/2n}} = \vert\vert x\vert\vert$. The basic models are the $ {L^{2n}}$ spaces, but one also has that every inner product space is a $ {G_{2n}}$ space for every integer $ n$. Hence $ {G_{2n}}$ spaces of a given cardinality need not be isometrically isomorphic. It is shown that a complex normed linear space is a $ {G_{2n}}$ space if and only if the norm satisfies a generalized parallelogram law. From the proof of this characterization it follows that a linear map $ U$ from $ X$ to $ X$ is an isometry if and only if $ \left\langle {U({x_1}), \ldots ,U({x_{2n}})} \right\rangle = \left\langle {{x_1}, \ldots ,{x_{2n}}} \right\rangle$ for all $ {x_1}, \ldots ,{x_{2n}}$. This then provides a way to construct all of the isometries of a finite dimensional $ {G_{2n}}$ space. Of particular interest are the $ \operatorname{CBS} {G_{2n}}$ spaces in which $ \vert\left\langle {{x_1}, \ldots ,{x_{2n}}} \right\rangle \vert \leqq \vert\vert{x_1}\vert\vert \cdots \vert\vert{x_{2n}}\vert\vert$. These spaces have many properties similar to inner product spaces. An operator $ A$ on a complete $ \operatorname{CBS} {G_{2n}}$ space is said to be symmetric if and only if $ \left\langle {{x_1}, \ldots ,A({x_i}), \ldots ,{x_{2n}}} \right\rangle = \left\langle {{x_1}, \ldots ,A({x_j}), \ldots ,{x_{2n}}} \right\rangle $ for all $ i$ and $ j$. It is easy to show that these operators are scalar and that on $ {L^{2n}},n > 1$, they characterize multiplication by a real $ {L^\infty }$ function. The interest in nontrivial symmetric operators is that they exist if and only if the space can be decomposed into the direct sum of nontrivial $ {G_{2n}}$ spaces.

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Keywords: Generalized-inner-products, semi-inner-products, inner products, norms from multilinear forms, generalized polarization identity, generalized parallelogram law, $ {G_{2n}}$ spaces, $ \operatorname{CBS} {G_{2n}}$ spaces, isometries on $ {G_{2n}}$ spaces, symmetric operators on $ \operatorname{CBS} {G_{2n}}$ spaces
Article copyright: © Copyright 1970 American Mathematical Society