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Boundedness and Invertibility

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

Abstract

Boundedness is a useful and natural condition, but it is a very strong condition on a linear transformation. The condition has a profound effect throughout operator theory, from its mildest algebraic aspects to its most complicated topological ones. To avoid certain obvious mistakes, it is important to know that boundedness is more than just the conjunction of an infinite number of conditions, one for each element of a basis. If A is an operator on a Hilbert space H with an orthonormal basis {e1, e2, e3,•••}, thenthenumbers ‖Aen‖ arebounded; if, forinstance, ‖A‖ ≦ 1, then ‖Aen‖ ≦ 1 for all n; and, of course, if A = 0, then Aen = 0 for all n. The obvious mistakes just mentioned are based on the assumption that the converses of these assertions are true.

Keywords

Hilbert Space Linear Transformation Uniform Boundedness Cardinal Number Linear Manifold 
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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