Analysis Now

1 General Topology.- 1.1. Ordered Sets.- The axiom of choice, Zorn's lemma, and Cantors's well-ordering principle; and their equivalence. Exercises..- 1.2. Topology.- Open and closed sets. Interior points and boundary. Basis and subbasis for a topology. Countability axioms. Exercises..- 1.3. Convergence.- Nets and subnets. Convergence of nets. Accumulation points. Universal nets. Exercises..- 1.4. Continuity.- Continuous functions. Open maps and homeomorphisms. Initial topology. Product topology. Final topology. Quotient topology. Exercises..- 1.5. Separation.- Hausdorff spaces. Normal spaces. Urysohn's lemma. Tietze's extension theorem. Semicontinuity. Exercises..- 1.6. Compactness.- Equivalent conditions for compactness. Normality of compact Hausdorff spaces. Images of compact sets. Tychonoff's theorem. Compact subsets of ?n. The Tychonoff cube and metrization. Exercises..- 1.7. Local Compactness.- One-point compactification. Continuous functions vanishing at infinity. Normality of locally compact, ?-compact spaces. Paracompactness. Partition of unity. Exercises..- 2 Banach Spaces.- 2.1. Normed Spaces.- Normed spaces. Bounded operators. Quotient norm. Finite-dimensional spaces. Completion. Examples. Sum and product of normed spaces. Exercises..- 2.2. Category.- The Baire category theorem. The open mapping theorem. The closed graph theorem. The principle of uniform boundedness. Exercises..- 2.3. Dual Spaces.- The Hahn-Banach extension theorem. Spaces in duality. Adjoint operator. Exercises..- 2.4. Seminormed Spaces.- Topological vector spaces. Seminormed spaces. Continuous functionals. The Hahn-Banach separation theorem. The weak* topology. w*-closed subspaces and their duality theory. Exercises..- 2.5. w*-Compactness.- Alaoglu's theorem. Krein-Milman's theorem. Examples of extremal sets. Extremal probability measures. Krein-Smulian's theorem. Vector-valued integration. Exercises..- 3 Hilbert Spaces.- 3.1. Inner Products.- Sesquilinear forms and inner products. Polarization identities and the Cauchy-Schwarz inequality. Parallellogram law. Orthogonal sum. Orthogonal complement. Conjugate self-duality of Hilbert spaces. Weak topology. Orthonormal basis. Orthonormalization. Isomorphism of Hilbert spaces. Exercises..- 3.2. Operators on Hilbert Space.- The correspondence between sesquilinear forms and operators. Adjoint operator and involution in B(?). Invertibility, normality, and positivity in B(?). The square root. Projections and diagonalizable operators. Unitary operators and partial isometries. Polar decomposition. The Russo-Dye-Gardner theorem. Numerical radius. Exercises..- 3.3. Compact Operators.- Equivalent characterizations of compact operators. The spectral theorem for normal, compact operators. Atkinson's theorem. Fredholm operators and index. Invariance properties of the index. Exercises..- 3.4. The Trace.- Definition and invariance properties of the trace. The trace class operators and the Hilbert-Schmidt operators. The dualities between B0(?), B1(?) and B(?). Fredholm equations. The Sturm-Liouville problem. Exercises..- 4 Spectral Theory.- 4.1. Banach Algebras.- Ideals and quotients. Unit and approximate units. Invertible elements. C. Neumann's series. Spectrum and spectral radius. The spectral radius formula. Mazur's theorem. Exercises..- 4.2. The Gelfand Transform.- Characters and maximal ideals. The Gelfand transform. Examples, including Fourier transforms. Exercises..- 4.3. Function Algebras.- The Stone-Weierstrass theorem. Involution in Banach algebras. C*-algebras. The characterization of commutative C*-algebras. Stone-Cech compactification of Tychonoff spaces. Exercises..- 4.4. The Spectral Theorem, I.- Spectral theory with continuous function calculus. Spectrum versus eigenvalues. Square root of a positive operator. The absolute value of an operator. Positive and negative parts of a self-adjoint operator. Fuglede's theorem. Regular equivalence of normal operators. Exercises..- 4.5. The Spectral Theorem, II.- Spectral theory with Borel function calculus. Spectral measures. Spectral projections and eigenvalues. Exercises..- 4.6. Operator Algebra.- Strong and weak topology on B(?). Characterization of strongly/weakly continuous functionals. The double commutant theorem. Von Neumann algebras. The ?-weak topology. The ?-weakly continuous functionals. The predual of a von Neumann algebra. Exercises..- 4.7. Maximal Commutative Algebras.- The condition Cyclic and separating vectors. ??(X) as multiplication operators. A measure-theoretic model for MAÇA's. Multiplicity-free operators. MAÇA's as a generalization of orthonormal bases. The spectral theorem revisited. Exercises..- 5 Unbounded Operators.- 5.1. Domains, Extensions, and Graphs.- Densely defined operators. The adjoint operator. Symmetric and self-adjoint operators. The operator T*T. Semibounded operators. The Friedrichs extension. Examples..- 5.2. The Cayley Transform.- The Cayley transform of a symmetric operator. The inverse transformation. Defect indices. Affiliated operators. Spectrum of unbounded operators..- 5.3. Unlimited Spectral Theory.- Normal operators affiliated with a MAÇA. The multiplicity-free case. The spectral theorem for an unbounded, self-adjoint operator. Stone's theorem. The polar decomposition..- 6 Integration Theory.- 6.1. Radon Integrals.- Upper and lower integral. Daniell's extension theorem. The vector lattice ?1(X). Lebesgue's theorems on monotone and dominated convergence. Stieltjes integrals..- 6.2. Measurability.- Sequentially complete function classes, ?-rings and ?-algebras. Borel sets and functions. Measurable sets and functions. Integrability of measurable functions..- 6.3. Measures.- Radon measures. Inner and outer regularity. The Riesz representation theorem. Essential integral. The ?-compact case. Extended integrability..- 6.4. LP-Aspaces.- Null functions and the almost everywhere terminology. The Hölder and Minkowski inequalities. Egoroff's theorem. Lusin's theorem. The Riesz-Fischer theorem. Approximation by continuous functions. Complex spaces. Interpolation between ?P-spaces..- 6.5. Duality Theory.- ?-compactness and ?-finiteness. Absolute continuity. The Radon-Nikodym theorem. Radon charges. Total variation. The Jordan decomposition. The duality between LP-spaces..- 6.6. Product Integrals.- Product integral. Fubini's theorem. Tonelli's theorem. Locally compact groups. Uniqueness of the Haar integral. The modular function. The convolution algebras L1(G) and M(G)..- List of Symbols.