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Boundedness and Invertibility

  • Paul R. Halmos
Part of the Graduate Texts in Mathematics book series (GTM, volume 19)

Abstract

Let {e1 e2, e3,…} be an orthonormal basis for a Hubert space H, and find a Hamel basis for H that contains each e n . Let f0 be an arbitrary but fixed element of that Hamel basis distinct from each e n (see Solution 7). A unique linear transformation A is defined on H by the requirement that Af 0 = f0 and Af = 0 for all other elements of the selected basis: in particular, Ae n = 0 for all n. If A were bounded, then its vanishing on each e n would imply that A = 0. This solves parts (a) and (b) of the problem.

Keywords

Hilbert Space Linear Transformation Borel Subset Cardinal Number Polar Decomposition 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag New York Inc. 1982

Authors and Affiliations

  • Paul R. Halmos

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