[ambient page * updated * 03 Jan '09]
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[ home ]
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[* garrett@umn.edu *]

- [ Distributions supported on hyperplanes
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*updated*10 May '08] ... Proof that distributions supported on hyperplanes are compositions of transverse differentiations with restriction and then evaluation against distributions on the hyperplanes. - [ Heisenberg's uncertainty inequality
]
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*updated*10 May '08] ... Proof of an inequality concerning Fourier transforms that has the interpretation traditionally ascribed to Heisenberg's uncertainty principle. - [ non-locally-convex topological
vector spaces ]
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*updated*10 May '08] ... Proof that ell-p spaces with 0 < p < 1 are not locally convex - [ Weak smoothness implies
strong smoothness ]
[
*updated*21 Nov '06] ... for functions f with values in a quasi-complete locally convex topological vectorspace V. That is, if the scalar-valued (Lf)(x) function is smooth for every continuous linear functional L on V, then the V-valued function f itself is smooth. (The present sense of "weak" does not directly refer to distributional derivatives.) - [ Uniqueness of invariant
distributions ]
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*updated*03 Aug '05] ...on Lie groups, totally disconnected groups, adele groups, etc.

** Course Notes: **

- [ Metric spaces ]
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*updated*30 Aug '05] Review of metric spaces. Baire category theorem, both for complete metric and locally compact Hausdorff spaces. - [ Spaces of functions ]
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*updated*16 Sep '08] Basic definitions and overview. Emphasis on common Banach spaces of k-times continuously differentiable functions. Introduces Frechet spaces. - (*) Review exercises, exercises on
function spaces
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*updated*03 Feb '06] - [ Hilbert spaces ]
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*updated*29 Mar '09] Basics. Cauchy-Schwarz-Bunyakovsky inequality. Convexity theorem. Orthogonality. Riesz-Fischer theorem. - (*) First exercises related to
Fourier series
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*updated*03 Feb '06] - [Banach spaces] [pdf] Basics of functional analysis: Banach-Steinhaus theorem (Uniform Boundedness), Open Mapping Theorem, Hahn-Banach Theorem, in the simple context of Banach spaces.
- [ Applications of
Banach space ideas to Fourier series ]
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*updated*19 Feb '05] Divergence of Fourier series of continuous functions. Riemann-Lebesgue lemma. Non-surjectivity of map from integrable periodic functions to sequences going to zero at infinity. - (*) Exercises related to Banach
spaces
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*updated*03 Feb '06] - [ operators on Hilbert spaces
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*updated*19 Feb '05] Continuity and boundedness, adjoints, eigenvalues, discrete/continuous/residual spectrum. - [ spectral theorem for
self-adjoint compact operators on Hilbert spaces ]
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*updated*18 Feb '12] - [ topological vector spaces ]
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*updated*25 Jul '11] General topological vector spaces, uniqueness of (Hausdorff) topology on finite-dimensional spaces. - [ Hahn-Banach theorems ]
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*updated*17 Jul '08] Basic results concerning locally convex topological vectorspaces: dominated extension theorem, separation theorem, corollaries. - [ categorical constructions
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*updated*09 Nov '10] Products, coproducts, projective limits, direct limits, treated as initial or final objects in suitable categories of diagrams, to give trivial proofs of uniqueness. Proofs by viewpoint. - (*) some exercises on
general topological vector spaces
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*updated*03 Feb '06] - [ vector-valued integrals
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*updated*18 Jul '11]*Quasi/local-completeness*as useful criterion for existence of Gelfand-Pettis (*weak*) integrals of continuous compactly-supported vector-valued functions. Proves quasi/local-completeness of most useful spaces, including test functions, spaces of linear maps, etc. - [
Banach-Alaoglu, variant Banach-Steinhaus, bipolars, weak-to-strong
principles
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*updated*16 Jul '08] - (*)
Exercises on weak topologies, integrals
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*updated*03 Feb '06] - (*)
Exercises on distributions
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*updated*03 Feb '06] - [
vector-valued holomorphic functions, weak-to-strong holomorphy
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*updated*19 Feb '05]

Miscellaneous related notes:

- Measurable choice functions
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*updated*19 Jan '13] After Dixmier: existence of measurable functions X->Y with graph inside a given measurable subset of the product X x Y. - Schwartz-function envelopes
of rapidly decreasing
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*updated*20 Feb '05] After Weil: rapidly decreasing functions can be dominated by Schwartz functions, Schwartz functions can be factored as products of two, etc. - [ alternation of roots ] ... property of solutions to certain eigenfunction problems
- [ Catalogue of topological vector spaces ] Some important and popular topological vectorspaces.
- [ Distribution-theoretic proof of Poisson Summation Formula ]
- [ A 'good spectral theorem'] This detailed outline sets up basic ideas about von Neumann algebras, direct integrals of Hilbert spaces, recovering more traditional (less illuminating) things like `resolutions of the identity' as artifacts.
- [ Bernstein's analytic continuation of complex powers of polynomials ]

Unless explicitly noted otherwise, everything here, work by Paul Garrett, is licensed under a Creative Commons Attribution 3.0 Unported License. ... [

The University of Minnesota explicitly requires that I state that