Functional Analysis 2016-17, MWF 1:25, Vincent 301
[ Dangerous and Illegal Operations
in Calculus ] ... intro to Schwartz' generalized
functions/distributions.
[ambient page updated ]
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[ garrett@math.umn.edu ]
Text will be notes posted here.
- 01 Overview of natural function spaces,
Gelfand-Pettis integrals, Levi-Sobolev spaces
- 02 Reviews of
- 03 Fourier series, functions and generalized functions on the circle
- 04 More general types of topological vector spaces
- 05 Fourier transforms, functions and generalized transforms on the line
- 06 Bounded operators on Hilbert spaces
- 07
- 08 More on bounded operators on Hilbert spaces
Further notes 2013-14:
Functional analysis 2012-13, MWF 1:25, Vincent 02
In reverse chronological order:
Older notes
- [ Intersections of opens, unions of
closeds, over compact families
]
[ updated ]
... In general, only finite intersections of opens are open, and only finite unions
of closeds are closed. However, in more structured situations the
same conclusions hold for compact families rather than finite.
- [ Riesz' Lemma
]
[ updated ]
... 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 Levi-Sobolev imbedding and
Rellich-Kondrachev lemma
]
[ updated ]
... simplest case of Levi-Sobolev imbedding: +1-index L^{2} Levi-Sobolev
space on [0,1] is inside continuous functions, and
Rellich-Kondrachev: the inclusion of +1-index Levi-Sobolev space into
L^{2}[0,1] is compact .
- [ Young's inequality (numerical case)
]
[ updated ]
... ab is less than a^{p}/p + b^{q}/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 ]
... 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 ]
... 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 ]
... 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 ]
... L^{2} differentiation, L^{2} Levi-Sobolev spaces, for
Fourier series and Fourier transforms.
- [ Compact operators on Banach spaces
]
[ updated ]
... 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 ]
... 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 ]
... 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 Levi-Sobolev spaces.
- [ uncountable coproducts
]
[ updated ]
... 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
L^{p}(I) with 0<p<1.
- [ compact unions of closed are closed,
compact intersections of open are open
]
[ updated ]
... In topological groups and in topological vector spaces...
- [ smoothing/mollifying
distributions
]
[ updated ]
... 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 ]
... 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 ]
... 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.
- [ Levi-Sobolev imbedding to Lipschitz spaces
]
[ updated ]
Slightly stronger Levi-Sobolev imbedding theorem, not merely addressing
continuous differentiability, but additional Lipschitz conditions on
highest derivatives.
- [ Unbounded operators, Friedrichs
extensions, resolvents
]
[ updated ]
- [ Peetre's theorem
]
[ updated ]
... A linear operator not increasing supports is a differential operator.
- [ Snake lemma, extensions,
Gamma function
]
[ updated ]
... Simple homological ideas prove unique extendability, illustrated
with homogeneous distributions and Gamma.
- [ Distributions supported on hyperplanes
]
[ updated ]
... 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 ]
... 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 ]
... Proof that ell-p spaces with 0 < p < 1 are not locally convex
- [ Weak smoothness implies
strong smoothness ]
[ updated
]
... 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 ]
...on Lie groups, totally disconnected groups, adele groups, etc.
Old Course Notes:
- [ Metric spaces ]
... [ updated ]
Review of metric spaces. Baire category theorem, both for
complete metric and locally compact Hausdorff spaces.
- [ Spaces of functions ]
... [ updated ]
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 ]
- [ Hilbert spaces ]
... [ updated ]
Basics. Cauchy-Schwarz-Bunyakovsky inequality. Convexity
theorem. Orthogonality. Riesz-Fischer theorem.
- (*) First exercises related to
Fourier series
... [ updated ]
- [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 ]
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 ]
- [ operators on Hilbert spaces
]
... [ updated ]
Continuity and boundedness, adjoints, eigenvalues,
discrete/continuous/residual spectrum.
- [ spectral theorem for
self-adjoint compact operators on Hilbert spaces ]
... [ updated ]
- [ topological vector spaces ]
... [ updated ]
General topological vector spaces, uniqueness of (Hausdorff) topology
on finite-dimensional spaces.
- [ Hahn-Banach theorems ]
... [ updated ]
Basic results concerning locally convex topological vectorspaces:
dominated extension theorem, separation theorem, corollaries.
- [ categorical constructions
]
... [ updated ]
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 ]
- [ vector-valued integrals
]
... [ updated ]
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 ]
- (*)
Exercises on weak topologies, integrals
... [ updated ]
- (*)
Exercises on distributions
... [ updated ]
- [
vector-valued holomorphic functions, weak-to-strong holomorphy
]
... [ updated ]
Miscellaneous old notes: