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 05 Jan '16]
...
[ home ]
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[ garrett@math.umn.edu ]
Continuing notes 201314, in reverse chronological order:
Functional analysis 201213, MWF 1:25, Vincent 02
In reverse chronological order:
 ...
 11 Unbounded operators on Hilbert spaces
 10 Fourier transforms, tempered distributions
 09 Schwartz' distributions
 08
 [
exercises 04 ]
[ updated 08 Feb '13]
 07
 [
exercises 03 ]
[ updated 10 Dec '12]
 06 Bounded operators on Hilbert spaces
 [ discussion
of exercises 02 ]
[ updated 01 Nov '12]
 05
 [
Banach spaces
]
[ updated 16 Mar '14] Riesz' lemma, corollaries of
Baire: BanachSteinhaus/uniformboundedness, open mapping, closed
graph; also, HahnBanach

[ application of
Banach space ideas to Fourier series
]
[ updated 25 Sep '12] basic negative results: nonconvergence of Fourier series of
continuous functions, nonsurjectivity of Fourier coefficient map to
the space of sequences going to 0 at infinity

[
exercises 02
] [ updated 09 Oct '12]
 04
 03
 02
[ Examples of function spaces
]
[ updated 15 Mar '14] Natural Banach spaces, (projective) limits of Banach spaces, Frechet spaces
 01
[ review of metric spaces
]
[ updated 02 Feb '14]
Older notes
 [ Intersections of opens, unions of
closeds, over compact families
]
[ updated 18 Aug '12]
... 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 04 Mar '12]
... that for nondense subspace X in Banach space Y, and for
0<r<1 there is y in Y with y=1 and inf xy< r, where the
inf is over x in X. Useful as a sort of Banachspace substitute for
orthogonality in Hilbert spaces, not hard to prove, but rarely labelled
by this name in texts, therefore oddly hard to find.
 [ Simplest LeviSobolev imbedding and
RellichKondrachev lemma
]
[ updated 20 Mar '12]
... simplest case of LeviSobolev imbedding: +1index L^{2} LeviSobolev
space on [0,1] is inside continuous functions, and
RellichKondrachev: the inclusion of +1index LeviSobolev space into
L^{2}[0,1] is compact .
 [ Young's inequality (numerical case)
]
[ updated 03 Mar '12]
... 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 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 selfadjointness
]
[ updated 23 Feb '12]
... criterion for uniqueness of selfadjoint extension of symmetric unbounded
operators. Cautionary examples of incomparable selfadjoint
extensions. Examples of symmetric operators with selfadjoint
closures.
 [ HilbertSchmidt, compact operators,
spectral theorem
]
[ updated 01 Mar '12]
... Spectral theorem for selfadjoint, compact
operators on Hilbert spaces. HilbertSchmidt operators are compact.
[Some really dumb cutandpaste errors corrected]
 [ Plancherel and spectral decompositions
]
[ updated 25 Mar '14]
... L^{2} differentiation, L^{2} LeviSobolev spaces, for
Fourier series and Fourier transforms.
 [ Compact operators on Banach spaces
]
[ updated 04 Mar '12]
... the basic FredholmRiesz theory of compact operators on Banach
spaces: nonzero spectrum consists entirely of eigenvalues,
eigenspaces are finitedimensional, the only accumulation point of
the spectrum is 0, and the Fredholm alternative: for
compact T and nonzero complex z,
either Tz 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]
... HilbertSchmidt operators on Hilbert spaces, simplest nuclear
Frechet spaces constructed as HilbertSchmidt limits of Hilbert
spaces, categorical tensor products, strong dual topologies and
colimits, Schwartz' kernel theorem for LeviSobolev 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
notnecessarilylocallyconvex topological vector spaces, basically
because of the existence of the specific notlocallyconvex spaces
L^{p}(I) with 0<p<1.
 [ compact unions of closed are closed,
compact intersections of open are open
]
[ updated 18 Aug '12]
... In topological groups and in topological vector spaces...
 [ smoothing/mollifying
distributions
]
[ updated 13 Mar '13]
... Using smooth approximate identities, arbitrary distributions are
approximated in the weak*topology by smooth
functions. GelfandPettis/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.
 [ LeviSobolev imbedding to Lipschitz spaces
]
[ updated 23 Nov '10]
Slightly stronger LeviSobolev imbedding theorem, not merely addressing
continuous differentiability, but additional Lipschitz conditions on
highest derivatives.
 [ Unbounded operators, Friedrichs
extensions, resolvents
]
[ updated 25 May '14]
 [ 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.
 [ nonlocallyconvex topological
vector spaces ]
[ updated 10 May '08]
... Proof that ellp 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 quasicomplete locally convex
topological vectorspace V. That is, if the scalarvalued (Lf)(x)
function is smooth for every continuous linear functional L on V, then
the Vvalued 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.
Old 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 ktimes continuously differentiable functions. Introduces Frechet spaces.
 (*) Review exercises, exercises on
function spaces
... [ updated 03 Feb '06]
 [ Hilbert spaces ]
... [ updated 29 Mar '09]
Basics. CauchySchwarzBunyakovsky inequality. Convexity
theorem. Orthogonality. RieszFischer theorem.
 (*) First exercises related to
Fourier series
... [ updated 03 Feb '06]
 [Banach spaces]
Basics of functional analysis: BanachSteinhaus theorem
(Uniform Boundedness), Open Mapping Theorem, HahnBanach 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. RiemannLebesgue
lemma. Nonsurjectivity 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
selfadjoint 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 finitedimensional spaces.
 [ HahnBanach 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]
 [ vectorvalued integrals
]
... [ updated 18 Jul '11]
Quasi/localcompleteness as useful criterion for existence of
GelfandPettis ( weak ) integrals of continuous
compactlysupported vectorvalued functions. Proves
quasi/localcompleteness of most useful spaces, including test
functions, spaces of linear maps, etc.
 [
BanachAlaoglu, variant BanachSteinhaus, bipolars, weaktostrong
principles
]
... [ updated 16 Jul '08]
 (*)
Exercises on weak topologies, integrals
... [ updated 03 Feb '06]
 (*)
Exercises on distributions
... [ updated 03 Feb '06]
 [
vectorvalued holomorphic functions, weaktostrong holomorphy
]
... [ updated 19 Feb '05]
Miscellaneous old notes:
Unless explicitly noted otherwise, everything here, work
by Paul Garrett, is licensed
under a Creative
Commons Attribution 3.0
Unported License.
...
[ garrett@math.umn.edu ]
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."