Banach Spaces
نویسنده
چکیده
Paul Garrett [email protected] http://www.math.umn.edu/ g̃arrett/ [This document is http://www.math.umn.edu/ ̃garrett/m/fun/notes 2012-13/05 banach.pdf] 1. Basic definitions 2. Riesz’ Lemma 3. Counter-example: non-existence of norm-minimizing element 4. Normed spaces of continuous linear maps 5. Dual spaces of normed spaces 6. Banach-Steinhaus/uniform-boundedness theorem 7. Open mapping theorem 8. Closed graph theorem 9. Hahn-Banach theorem We have seen that many interesting spaces of functions, such as C(K) for K compact, and C[a, b], have natural structures of Banach spaces. Abstractly, Banach spaces are less convenient than Hilbert spaces, but still sufficiently simple so many important properties hold. Several standard results true in greater generality have simpler proofs for Banach spaces. Riesz’ lemma is an elementary result often an adequate substitute in Banach spaces for the lack of sharper Hilbert-space properties. We include a natural counter-example in the Banach space C[a, b] to the minimum principle valid in Hilbert spaces, but not generally valid in Banach spaces. The Banach-Steinhaus/uniform-boundedness theorem, open mapping theorem, and closed graph theorem are not elementary, since they invoke the Baire category theorem. The Hahn-Banach theorem is non-trivial, but does not use completeness.
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