File : pdf, 1.5 MB, 393 pages
by Shlomo Sternberg
TOC
1 The topology of metric spaces
1.1 Metric spaces
1.2 Completeness and completion
1.3 Normed vector spaces and Banach spaces
1.4 Compactness
1.5 Total Boundedness
1.6 Separability
1.7 Second Countability
1.8 Conclusion of the proof of Theorem
1.9 Dini’s lemma
1.10 The Lebesgue outer measure of an interval is its length
1.11 Zorn’s lemma and the axiom of choice
1.12 The Baire category theorem
1.13 Tychonoff’s theorem
1.14 Urysohn’s lemma
1.15 The Stone-Weierstrass theorem
1.16 Machado’s theorem
1.17 The Hahn-Banach theorem
1.18 The Uniform Boundedness Principle
2 Hilbert Spaces and Compact operators
2.1 Hilbert space
2.2 Self-adjoint transformations
2.3 Compact self-adjoint transformations
2.4 Fourier’s Fourier series
2.5 The Heisenberg uncertainty principle
2.6 The Sobolev Spaces
2.7 Garding’s inequality
2.8 Consequences of Garding’s inequality
2.9 Extension of the basic lemmas to manifolds
2.10 Example: Hodge theory
2.11 The resolvent
3 The Fourier Transform
3.1 Conventions, especially about 2phi
3.2 Convolution goes to multiplication
3.3 Scaling
3.4 Fourier transform of a Gaussian is a Gaussian
3.5 The multiplication formula
3.6 The inversion formula
3.7 Plancherel’s theorem
3.8 The Poisson summation formula
3.9 The Shannon sampling theorem
3.10 The Heisenberg Uncertainty Principle
3.11 Tempered distributions
4 Measure theory
4.1 Lebesgue outer measure
4.2 Lebesgue inner measure
4.3 Lebesgue’s definition of measurability
4.4 Caratheodory’s definition of measurability
4.5 Countable additivity
4.6 -fields, measures, and outer measures
4.7 Constructing outer measures, Method I
4.8 Constructing outer measures, Method II
4.9 Hausdorff measure
4.10 Hausdorff dimension
4.11 Push forward
4.12 The Hausdorff dimension of fractals
4.13 The Hausdorff metric and Hutchinson’s theorem
4.14 Affine examples
5 The Lebesgue integral
5.1 Real valued measurable functions
5.2 The integral of a non-negative function
5.3 Fatou’s lemma
5.4 The monotone convergence theorem
5.5 The space L1(X,R)
5.6 The dominated convergence theorem
5.7 Riemann integrability
5.8 The Beppo – Levi theorem
5.9 L1 is complete
5.10 Dense subsets of L1(R,R)
5.11 The Riemann-Lebesgue Lemma
5.12 Fubini’s theorem
6 The Daniell integral
6.1 The Daniell Integral
6.2 Monotone class theorems
6.3 Measure
6.4 Holder, Minkowski , Lp and Lq
6.5 the essential sup norm
6.6 The Radon-Nikodym Theorem
6.7 The dual space of Lp
6.8 Integration on locally compact Hausdorff spaces
6.8.1 Riesz representation theorems
6.8.2 Fubini’s theorem
6.9 The Riesz representation theorem redux
6.10 Existence
7 Wiener measure, Brownian motion and white noise
7.1 Wiener measure
7.2 Stochastic processes and generalized stochastic processes
7.3 Gaussian measures
7.3.1 Generalities about expectation and variance
7.3.2 Gaussian measures and their variances
7.3.3 The variance of a Gaussian with density
7.3.4 The variance of Brownian motion.
7.4 The derivative of Brownian motion is white noise.
8 Haar measure
8.1 Examples
8.2 Topological facts
8.3 Construction of the Haar integral
8.4 Uniqueness
8.5 μ(G) < 1 if and only if G is compact
8.6 The group algebra
8.7 The involution
8.8 The algebra of finite measures
8.9 Invariant and relatively invariant measures on homogeneous spaces
9 Banach algebras and the spectral theorem
9.1 Maximal ideals
9.2 Normed algebras
9.3 The Gelfand representation
9.3.1 Invertible elements in a Banach algebra form an open set
9.3.2 The Gelfand representation for commutative Banach algebras
9.3.3 The spectral radius
9.3.4 The generalized Wiener theorem
9.4 Self-adjoint algebras
9.5 The Spectral Theorem for Bounded Normal Operators, Functional Calculus Form
10 The spectral theorem
10.1 Resolutions of the identity
10.2 The spectral theorem for bounded normal operators
10.3 Stone’s formula
10.4 Unbounded operators
10.5 Operators and their domains
10.6 The adjoint
10.7 Self-adjoint operators
10.8 The resolvent
10.9 The multiplication operator form of the spectral theorem
10.10 The Riesz-Dunford calculus.
10.11 Lorch’s proof of the spectral theorem
10.12 Characterizing operators with purely continuous spectrum
10.13 Appendix. The closed graph theorem
11 Stone’s theorem
11.1 von Neumann’s Cayley transform
11.2 Equibounded semi-groups on a Frechet space
11.3 The differential equation
11.4 The power series expansion of the exponential
11.5 The Hille Yosida theorem
11.6 Contraction semigroups
11.7 Convergence of semigroups
11.8 The Trotter product formula
11.9 The Feynman-Kac formula
11.10 The free Hamiltonian and the Yukawa potential
12 More about the spectral theorem
12.1 Bound states and scattering states
12.2 Non-negative operators and quadratic forms
12.2.1 Fractional powers of a non-negative self-adjoint operator
12.2.2 Quadratic forms
12.2.3 Lower semi-continuous functions
12.2.4 The main theorem about quadratic forms
12.2.5 Extensions and cores
12.2.6 The Friedrichs extension
12.3 Dirichlet boundary conditions
12.4 Rayleigh-Ritz and its applications
12.5 The Dirichlet problem for bounded domains
12.6 Valence
12.7 Davies’s proof of the spectral theorem
13 Scattering theory
13.1 Examples
13.1.1 Translation – truncation
13.1.2 Incoming representations
13.1.3 Scattering residue
13.2 Breit-Wigner
13.3 The representation theorem for strongly contractive semi-groups
13.4 The Sinai representation theorem
13.5 The Stone – von Neumann theorem
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