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15 - We Go Deeper, DeeperWe Go (into the Structure of Complete Spaces)
Adam Bobrowski, Politechnika Lubelska, Poland
Book:

Functional Analysis Revisited

Published online:

31 October 2024

Print publication:

04 July 2024, pp 185-209
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Summary

The celebrated Baire’s category theorem says that a complete space cannot be represented as a countable union of nowhere dense sets. This is a fundamental description of the structure of complete spaces. Because of this, it is fitting to derive the Banach–Steinhaus theorem as a consequence of Baire’s. This is what we do at the beginning of this chapter. We also show that the set of differentiable functions is quite small (i.e. meagre) in the space of continuous functions. As further consequences of Baire’s theorem we discuss two other fundamental results of functional analysis – the open mapping theorem and the closed graph theorem – together with some of their most immediate applications. In the meantime, we use the Banach–Steinhaus theorem to show that a Fourier series cannot converge uniformly for all continuous (and periodic) functions.

On the Adjoint and the Closure of the Sum of Two Unbounded Operators
Mohammed Hichem Mortad
Journal:

Canadian Mathematical Bulletin / Volume 54 / Issue 3 / 01 September 2011

Published online by Cambridge University Press:

20 November 2018, pp. 498-505

Print publication:

01 September 2011
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We prove, under some conditions on the domains, that the adjoint of the sum of two unbounded operators is the sum of their adjoints in both Hilbert and Banach space settings. A similar result about the closure of operators is also proved. Some interesting consequences and examples “spice up” the paper.

On Operators with Spectral Square but without Resolvent Points
Paul Binding, Vladimir Strauss
Journal:

Canadian Journal of Mathematics / Volume 57 / Issue 1 / 01 February 2005

Published online by Cambridge University Press:

20 November 2018, pp. 61-81

Print publication:

01 February 2005
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Decompositions of spectral type are obtained for closed Hilbert space operators with empty resolvent set, but whose square has closure which is spectral. Krein space situations are also discussed.