Functional Analysis
See also
[vignettes],
[intro mfms],
[Lie theory],
[algebra],
[complex analysis],
[ Dangerous and Illegal Operations
in Calculus ] ... intro to Schwartz' generalized
functions/distributions.
[ambient page updated 14 May '12]
...
[ home ]
...
[ garrett@math.umn.edu ]
Functional analysis 2012-13, MWF 1:25, Vincent 02
Text will be a suitable rearrangement of notes below. Also, there
are several reasonable texts that are useful supplements.
The course will be driven by examples and applications to harmonic
analysis, differential equations, representation theory, and number
theory. Sample discussions, which will be updated and integrated:
- [ Riesz' Lemma
]
[ updated 04 Mar '12]
... that for non-dense subspace X in Banach space Y, and for
0<r<1 there is y in Y with |y|=1 and inf |x-y|< r, where the
inf is over x in X. Useful as a sort of Banach-space substitute for
orthogonality in Hilbert spaces, not hard to prove, but rarely labelled
by this name in texts, therefore oddly hard to find.
- [ Simplest Sobolev imbedding and
Rellich-Kondrachev lemma
]
[ updated 20 Mar '12]
... simplest case of Sobolev imbedding: +1-index L2 Sobolev
space on [0,1] is inside continuous functions, and
Rellich-Kondrachev: the inclusion of +1-index Sobolev space into
L2[0,1] is compact .
- [ Young's inequality (numerical case)
]
[ updated 03 Mar '12]
... ab is less than ap/p + bq/q for conjugate exponents
1/p+1/q=1. It's
easy enough, just convexity of log, but p=q=2 is even easier,
sometimes making the general case mysteriously difficult by
comparison. Young's 1912 papers cited (locations retrieved from Wiki).
- [ Simple example Friedrichs extensions
of restrictions of Laplacians
]
[ updated 01 Mar '12]
... giving spectral decomposition of Laplacian on [a,b] from that of
the whole line. Obtaining Dirichlet problem boundary condition at
endpoints.
- [ Criterion for essential self-adjointness
]
[ updated 23 Feb '12]
... criterion for uniqueness of self-adjoint extension of symmetric unbounded
operators. Cautionary examples of incomparable self-adjoint
extensions. Examples of symmetric operators with self-adjoint
closures.
- [ Hilbert-Schmidt, compact operators,
spectral theorem
]
[ updated 01 Mar '12]
... Spectral theorem for self-adjoint, compact
operators on Hilbert spaces. Hilbert-Schmidt operators are compact.
[Some really dumb cut-and-paste errors corrected]
- [ Plancherel and spectral decompositions
]
[ updated 26 Feb '12]
... L2 differentiation, L2 Sobolev spaces, for
Fourier series and Fourier transforms.
- [ Compact operators on Banach spaces
]
[ updated 04 Mar '12]
... the basic Fredholm-Riesz theory of compact operators on Banach
spaces: non-zero spectrum consists entirely of eigenvalues,
eigenspaces are finite-dimensional, the only accumulation point of
the spectrum is 0, and the Fredholm alternative: for
compact T and nonzero complex z,
either T-z is a bijection, or
its kernel and cokernel have the same (finite) dimension (and the
image is closed).
- [ Compact resolvents and perturbations
]
[ updated 24 Jul '11]
... Especially for unbounded operators on Hilbert or Banach spaces,
compactness of the resolvent, and the pursuant meromorphy in the
spectral parameter is very important. We prove that compactness of
the resolvent at any single point implies meromorphy and compactness
everywhere away from poles.
- [ nuclear spaces and kernel
theorem I
]
[ updated 19 Jul '11]
... Hilbert-Schmidt operators on Hilbert spaces, simplest nuclear
Frechet spaces constructed as Hilbert-Schmidt limits of Hilbert
spaces, categorical tensor products, strong dual topologies and
colimits, Schwartz' kernel theorem for Sobolev spaces.
- [ uncountable coproducts
]
[ updated 18 Jul '11]
... of locally convex topological vector spaces, in the locally convex
category, fail to be coproducts in the larger category of
not-necessarily-locally-convex topological vector spaces, basically
because of the existence of the specific not-locally-convex spaces
Lp(I) with 0<p<1.
- [ compact unions of closed are closed,
compact intersections of open are open
]
[ updated 28 Jun '11]
... In topological groups and in topological vector spaces...
- [ smoothing/mollifying
distributions
]
[ updated 05 Apr '12]
... Using smooth approximate identities, arbitrary distributions are
approximated in the weak topology by smooth
functions. Gelfand-Pettis (weak) integrals play a central role.
- [ weak duals are not complete
]
[ updated 02 Jan '11]
... Weak duals of reasonable topological vector spaces are not
complete. This has been known since 1950 work of
Grothendieck. Fortunately, quasi- completeness is
sufficient in practice. Sequential completeness is
insufficient.
- [ Distributions supported at 0
]
[ updated 16 Dec '10]
... The primordial result that distributions supported at 0 are finite
linear combinations of Dirac delta and its partial
derivatives. Known long before the notion of distribution was made
explicit.
- [ Sobolev imbedding to Lipschitz spaces
]
[ updated 23 Nov '10]
Slightly stronger Sobolev imbedding theorem, not merely addressing
continuous differentiability, but additional Lipschitz conditions on
highest derivatives.
- [ Unbounded operators, Friedrichs
extensions, resolvents
]
[ updated 07 Mar '11]
- [ Peetre's theorem
]
[ updated 16 Oct '09]
... A linear operator not increasing supports is a differential operator.
- [ Snake lemma, extensions,
Gamma function
]
[ updated 14 Jun '11]
... Simple homological ideas prove unique extendability, illustrated
with homogeneous distributions and Gamma.
- [ 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 ]
... [ updated 30 Aug '05]
Review of metric spaces. Baire category theorem, both for
complete metric and locally compact Hausdorff spaces.
- [ Spaces of functions ]
... [ 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
... [ updated 03 Feb '06]
- [ Hilbert spaces ]
... [ updated 29 Mar '09]
Basics. Cauchy-Schwarz-Bunyakovsky inequality. Convexity
theorem. Orthogonality. Riesz-Fischer theorem.
- (*) First exercises related to
Fourier series
... [ updated 03 Feb '06]
- [Banach spaces]
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 ]
... [ 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
... [ updated 03 Feb '06]
- [ operators on Hilbert spaces
]
... [ updated 19 Feb '05]
Continuity and boundedness, adjoints, eigenvalues,
discrete/continuous/residual spectrum.
- [ spectral theorem for
self-adjoint compact operators on Hilbert spaces ]
... [ updated 18 Feb '12]
- [ topological vector spaces ]
... [ updated 25 Jul '11]
General topological vector spaces, uniqueness of (Hausdorff) topology
on finite-dimensional spaces.
- [ Hahn-Banach theorems ]
... [ updated 17 Jul '08]
Basic results concerning locally convex topological vectorspaces:
dominated extension theorem, separation theorem, corollaries.
- [ categorical constructions
]
... [ 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
[ updated 03 Feb '06]
- [ vector-valued integrals
]
... [ 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
]
... [ updated 16 Jul '08]
- (*)
Exercises on weak topologies, integrals
... [ updated 03 Feb '06]
- (*)
Exercises on distributions
... [ updated 03 Feb '06]
- [
vector-valued holomorphic functions, weak-to-strong holomorphy
]
... [ updated 19 Feb '05]
Miscellaneous old notes:
© 1996-2011, Creative Commons license,
This work
by Paul Garrett is licensed
under a Creative
Commons Attribution 3.0
Unported License.
...
[ garrett@math.umn.edu ]
[this page is http://www.math.umn.edu/~garrett/m/fun/]
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