Functional Analysis 2016-17, MWF 1:25, Vincent 364

See also [vignettes], [intro mfms], [Lie theory], [algebra], [complex analysis], [real analysis], [repn theory], [buildings notes]

[ Dangerous and Illegal Operations in Calculus ] ... intro to Schwartz' generalized functions/distributions.

[ambient page updated 28 Nov '16] ... [ home ] ... [ garrett@math.umn.edu ]

Text will be notes posted here.

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 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 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 20 Mar '12] ... simplest case of Levi-Sobolev imbedding: +1-index L2 Levi-Sobolev space on [0,1] is inside continuous functions, and Rellich-Kondrachev: the inclusion of +1-index Levi-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 25 Mar '14] ... L2 differentiation, L2 Levi-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 Levi-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 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. 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.
• [ Levi-Sobolev imbedding to Lipschitz spaces ] [ updated 23 Nov '10] Slightly stronger Levi-Sobolev 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.
• [ 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.

Old Course Notes:

Miscellaneous old notes: