Teschl Gerald. Topics in Real and Functional Analysis

Universitaet Wien, 2016. — 486 p.This manuscript provides a brief introduction to Real and (linear and nonlinear) Functional Analysis. It covers basic Hilbert and Banach space theory as well as basic measure theory including Lebesgue spaces and the Fourier transform.Functional AnalysisWarm up: Metric and topological spacesBasics
Convergence and completeness
Functions
Product topologies
Compactness
Separation
Connectedness
Continuous functions on metric spacesA first look at Banach and Hilbert spacesIntroduction: Linear partial diferential equations
The Banach space of continuous functions
The geometry of Hilbert spaces
Completeness
Bounded operators
Sums and quotients of Banach spaces
Spaces of continuous and diferentiable functionsHilbert spacesOrthonormal bases
The projection theorem and the Riesz lemma
Operators de_ned via forms
Orthogonal sums and tensor products
Applications to Fourier seriesCompact operatorsCompact operators
The spectral theorem for compact symmetric operators
Applications to Sturm{Liouville operators
Estimating eigenvaluesThe main theorems about Banach spacesThe Baire theorem and its consequences
The Hahn{Banach theorem and its consequences
The adjoint operator
The geometric Hahn{Banach theorem
Weak convergence
Weak topologies
Beyond Banach spaces: Locally convex spacesMore on compact operatorsCanonical form of compact operators
Hilbert{Schmidt and trace class operators
Fredholm theory for compact operatorsBounded linear operatorsBanach algebras
The C_ algebra of operators and the spectral theorem
Spectral measures
The Stone{Weierstra_ theorem
The Gelfand representation theoremOperator semigroupsAnalysis for Banach space valued functions
Uniformly continuous operator groups
Strongly continuous semigroups
Generator theoremsReal AnalysisMeasuresThe problem of measuring sets
Sigma algebras and measures
Extending a premeasure to a measure
Borel measures
Measurable functions
How wild are measurable objects
Appendix: Jordan measurable sets
Appendix: Equivalent de_nitions for the outer Lebesgue measureIntegrationIntegration | Sum me up, Henri
Product measures
Transformation of measures and integrals
Appendix: Transformation of Lebesgue{Stieltjes integrals
Appendix: The connection with the Riemann integralThe Lebesgue spaces LpFunctions almost everywhere
Jensen _ Holder _ Minkowski
Nothing missing in Lp
Approximation by nicer functions
Integral operatorsMore measure theoryDecomposition of measures
Derivatives of measures
Complex measures
Hausdor_ measure
In_nite product measures
The Bochner integral
Weak and vague convergence of measures
Appendix: Functions of bounded variation and absolutely continuous functionsThe dual of LpThe dual of Lp, p <
The dual of L and the Riesz representation theorem
The Riesz{Markov representation theoremThe Fourier transformThe Fourier transform on L and L
Applications to linear partial diferential equations
Sobolev spaces
Applications to evolution equations
Tempered distributionsInterpolationInterpolation and the Fourier transform on Lp
The Marcinkiewicz interpolation theoremNonlinear Functional AnalysisAnalysis in Banach spacesDiferentiation and integration in Banach spaces
Contraction principles
Ordinary diferential equationsThe Brouwer mapping degreeDe_nition of the mapping degree and the determinant formula
Extension of the determinant formula
The Brouwer _xed-point theorem
Kakutani's _xed-point theorem and applications to game theory
Further properties of the degree
The Jordan curve theoremThe Leray{Schauder mapping degreeThe mapping degree on _nite dimensional Banach spaces
Compact maps
The Leray{Schauder mapping degree
The Leray{Schauder principle and the Schauder _xed-point theorem
Applications to integral and diferential equationsThe stationary Navier{Stokes equationIntroduction and motivation
An insert on Sobolev spaces
Existence and uniqueness of solutionsMonotone mapsMonotone maps
The nonlinear Lax{Milgram theorem
The main theorem of monotone mapsAppendix A Some set theoryGlossary of notation