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The analytic properties of $ G\sb{2n}$ spaces


Author: Donald O. Koehler
Journal: Proc. Amer. Math. Soc. 35 (1972), 201-206
MSC: Primary 46B05
DOI: https://doi.org/10.1090/S0002-9939-1972-0301486-6
MathSciNet review: 0301486
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Abstract: A complex vector space X will be called an $ {F_{2n}}$ space if and only if there is a mapping $ \langle \cdot , \cdots , \cdot \rangle $ from $ {X^{2n}}$ into the complex numbers such that: $ \langle x, \cdots ,x\rangle > 0$ if $ x \ne 0;{x_k} \to \langle {x_1}, \cdots ,{x_{2n}}\rangle $ is linear for $ k = 1, \cdots ,n;\langle {x_1}, \cdots ,{x_{2n}}\rangle = {\langle {x_{2n}}, \cdots ,{x_1}\rangle ^ - }$ where denotes complex conjugate; $ \langle {x_{\sigma (1)}}, \cdots ,{x_{\sigma (n)}},{y_{\tau (1)}}, \cdots ,{y_{\tau (n)}}\rangle = \langle {x_1}, \cdots ,{x_n},{y_1}, \cdots ,{y_n}\rangle $ for all permutations $ \sigma ,\tau $ of $ \{ 1, \cdots ,n\} $. In the case of a real vector space the mapping is assumed to be into the reals such that: $ \langle x, \cdots ,x\rangle > 0$ if $ x \ne 0;{x_k} \to \langle {x_1}, \cdots ,{x_{2n}}\rangle $ is linear for $ k = 1, \cdots ,2n;\langle {x_{\sigma (1)}}, \cdots ,{x_{\sigma (2n)}}\rangle = \langle {x_1}, \cdots ,{x_{2n}}\rangle $ for all permutations $ \sigma $ of $ \{ 1, \cdots ,2n\} $. In either case, if $ \left\Vert x \right\Vert = {\langle x, \cdots ,x\rangle ^{1/2n}}$ defines a norm, X is called a $ {G_{2n}}$ space (Trans. Amer. Math. Soc. 150 (1970), 507-518). It is shown that an $ {F_{2n}}$ space is a $ {G_{2n}}$ space if and only if $ \vert\langle x,y, \cdots ,y\rangle {\vert^{2n}} \leqq \langle x, \cdots ,x\rangle {\langle y, \cdots ,y\rangle ^{2n - 1}}$ and that $ {G_{2n}}$ spaces are examples of uniform semi-inner-product spaces studied by Giles (Trans. Amer. Math. Soc. 129 (1967), 436-446).


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DOI: https://doi.org/10.1090/S0002-9939-1972-0301486-6
Keywords: Uniform semi-inner-product spaces, $ {G_{2n}}$ spaces, $ {F_{2n}}$ spaces, norms from multilinear forms, operators preserving multilinear forms
Article copyright: © Copyright 1972 American Mathematical Society

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