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Proceedings of the American Mathematical Society

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

 

 

On spaces with norms of negative and positive order


Author: Gideon Peyser
Journal: Proc. Amer. Math. Soc. 33 (1972), 81-88
MSC: Primary 46C05
DOI: https://doi.org/10.1090/S0002-9939-1972-0296665-0
MathSciNet review: 0296665
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Abstract: The two Hilbert spaces $ {H_0}$ and $ {H_1}$ are defined to be a generating pair if $ {H_1}$ is a dense subspace of $ {H_0}$ and if the norm of an element in $ {H_1}$ is greater than or equal to the norm in $ {H_0}$. It is shown that the pair generates a sequence of spaces $ \{ {H_k}\}, - \infty < k < \infty $, such that any two spaces of the sequence form again a generating pair. Such a pair is shown to generate, in turn, a subsequence of $ \{ {H_k}\} $. Also, representation theorems are derived for bounded linear functionals over the spaces of the sequence $ \{ {H_k}\} $, generalizing the Lax representation theorem and the Lax-Milgram theorem.


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DOI: https://doi.org/10.1090/S0002-9939-1972-0296665-0
Keywords: Hilbert space, norm, inner product, weaker space, stronger space, lower space, upper space, isometric isomorphism, correspondence, bounded linear functional, representation theorem
Article copyright: © Copyright 1972 American Mathematical Society