Functional Analysis
[ Dangerous and Illegal Operations
in Calculus ] [pdf]... intro to Schwartz'
generalized functions/distributions.
[ambient page updated 10 May '08]
...
[ home ]
...
[ garrett@math.umn.edu ]
- [ Distributions supported on hyperplanes
]
[ 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
]
[ 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 ]
[ 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 ]
[ updated 03 Aug '05]
...on Lie groups, totally disconnected groups, adele groups, etc.
Course Notes:
- [ Metric spaces ]
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[pdf]
... [ 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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[pdf]
... [ updated 29 Aug '05]
Basic definitions and overview. Emphasis on common Banach spaces
of k-times continuously differentiable functions. Introduces Frechet spaces.
- (*) Review exercises, exercises on
function spaces
... [ updated 31 Oct '03]
- [ Hilbert spaces ]
[pdf]
... [ updated 21 Apr '07]
Basics. Cauchy-Schwarz-Bunyakovsky inequality. Convexity
theorem. Orthogonality. Riesz-Fischer theorem.
- (*) First exercises related to
Fourier series
... [ updated 31 Oct '03]
- [Banach spaces]
[pdf]
... [ updated 30 Aug '05]
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 ]
[pdf]
... [ updated 31 Dec '04]
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
... [ updated 05 Nov '03]
- [ operators on Hilbert spaces
]
[pdf]
... [ updated 31 Dec '04]
Continuity and boundedness, adjoints, eigenvalues,
discrete/continuous/residual spectrum.
- [ spectral theorem for
self-adjoint compact operators on Hilbert spaces ]
[pdf]
... [ updated 31 Dec '04]
- [ topological vector spaces ]
[pdf]
... [ updated 30 Aug '05]
General topological vector spaces, uniqueness of (Hausdorff) topology
on finite-dimensional spaces.
- [ categorical constructions
]
[pdf]
... [ updated 10 Feb '06]
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
[ updated 08 Mar '04]
- [ vector-valued integrals
]
[pdf]
... [ updated 24 Apr '07]
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
]
[pdf]
... [ updated 11 Jan '07]
- (*)
Exercises on weak topologies, integrals
... [ updated 12 Apr '04]
- (*)
Exercises on distributions
... [ updated 12 Apr '04]
- [
vector-valued holomorphic functions, weak-to-strong holomorphy
]
[pdf]
... [ updated 12 Apr '04]
Miscellaneous related notes:
© 1996-2008,
Paul Garrett
...
[ garrett@math.umn.edu
]
[this page is http://www.math.umn.edu/~garrett/m/fun/]
The University of Minnesota explicitly requires that I
state that "The views and opinions expressed in this page are
strictly those of the page author. The contents of this page have not
been reviewed or approved by the University of Minnesota."