spaces

Author:
Donald O. Koehler

Journal:
Trans. Amer. Math. Soc. **150** (1970), 507-518

MSC:
Primary 46.15; Secondary 47.00

DOI:
https://doi.org/10.1090/S0002-9947-1970-0262806-7

MathSciNet review:
0262806

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Abstract | References | Similar Articles | Additional Information

Abstract: A complex normed linear space will be called a *space* if and only if there is a mapping from into the complex numbers such that: is linear for ; and . The basic models are the spaces, but one also has that every inner product space is a space for every integer . Hence spaces of a given cardinality need not be isometrically isomorphic. It is shown that a complex normed linear space is a space if and only if the norm satisfies a generalized parallelogram law. From the proof of this characterization it follows that a linear map from to is an isometry if and only if for all . This then provides a way to construct all of the isometries of a finite dimensional space. Of particular interest are the spaces in which . These spaces have many properties similar to inner product spaces. An operator on a complete space is said to be symmetric if and only if for all and . It is easy to show that these operators are scalar and that on , they characterize multiplication by a real 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 spaces.

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1970-0262806-7

Keywords:
Generalized-inner-products,
semi-inner-products,
inner products,
norms from multilinear forms,
generalized polarization identity,
generalized parallelogram law,
spaces,
spaces,
isometries on spaces,
symmetric operators on spaces

Article copyright:
© Copyright 1970
American Mathematical Society