Before approaching this material, students should have a familiarity with the basics of real variable theory and point set topology. This graduate-level mathematics textbook was used in the instruction of real variables and functional analysis. It was written in 2005 by Shlomo Sternberg of Harvard University.

*Theory of Functions of a Real Variable* is somewhat broken down into two main sections – 1. measure theory and integration and 2. Hilbert space theory, particularly the spectral theorem and its applications.

In Sternberg’s table of contents, he breaks down the topics to the point where it rivals an index. A great opportunity for students visiting The Free Textbook List to find help with specific topics and supplement their assigned course materials. If you find this material helpful, you might also want to check out two other mathematics textbooks he has written… Semi-Riemann Geometry and General Relativity and Lie Algebras.

## Table of Contents

- The topology of metric spaces
- Metric spaces
- Completeness and completion
- Normed vector spaces and Banach spaces
- Compactness
- Total Boundedness
- Separability
- Second Countability
- Conclusion of the proof of Theorem
- Dini’s lemma
- The Lebesgue outer measure of an interval is its length
- Zorn’s lemma and the axiom of choice
- The Baire category theorem
- Tychonoff’s theorem
- Urysohn’s lemma
- The Stone-Weierstrass theorem
- Machado’s theorem
- The Hahn-Banach theorem
- The Uniform Boundedness Principle

- Hilbert Spaces and Compact operators
- Hilbert space
- Scalar products
- The Cauchy-Schwartz inequality
- The triangle inequality
- Hilbert and pre-Hilbert spaces
- The Pythagorean theorem
- The theorem of Apollonius
- The theorem of Jordan and von Neumann
- Orthogonal projection
- The Riesz representation theorem
- What is L(T)?
- Projection onto a direct sum
- Projection onto a finite dimensional subspace
- Bessel’s inequality
- Parseval’s equation
- Orthonormal bases

- Self-adjoint transformations
- Non-negative self-adjoint transformations

- Compact self-adjoint transformations
- Fourier’s Fourier series
- Proof by integration by parts
- Relation to the operator
- Girding’s inequality, special case

- The Heisenberg uncertainty principle
- The Sobolev Spaces
- G Arding’s inequality
- Consequences of Girding’s inequality
- Extension of the basic lemmas to manifolds
- Example: Hodge Theory
- The resolvant

- Hilbert space
- The Fourier Transform
- Conventions, especially about r
- Convolution goes to multiplication
- Scaling
- Fourier transform of a Gaussian is a Gaussian
- The multiplication formula
- The inversion formula
- Plancherel’s theorem
- The Poisson summation formula
- The Shannon sampling theorem
- The Heisenberg Uncertainty Principle
- Tempered distributions
- Examples of Fourier transforms of elements of S’

- Measure Theory
- Lebesgue outer measure
- Lebesgue inner measure
- Lebesgue’s definition of measurability
- Caratheodory’s definition of measurability
- Countable additivity
- o-fields, measures, and outer measures
- Constructing outer measures, Method I
- A pathological example
- Metric outer measures

- Constructing outer measures, Method II
- Hausdorff measure
- Hausdorff dimension
- Push forward
- The Hausdorff dimension of fractals
- Similarity dimension
- The string model

- The Hausdorff metric and Hutchinson’s theorem
- Affine examples
- The classical Cantor set
- The Sierpinski Gasket
- Moran’s theorem

- The Lebesgue integral
- Real valued measurable functions
- The integral of a non-negative function
- Fatou’s lemma
- The monotone convergence theorem
- The space L(X, R)
- The dominated convergence theorem
- Riemann integrability
- The Beppo – Levi theorem
- L1 is complete
- Dense subsets of L1 (R, R)
- The Riemann-Lebesgue Lemma
- The Cantor-Lebesgue theorem

- Fubini’s theorem
- Product o–fields
- pi-systems and lamda-system
- The monotone class theorem
- Fubini for finite measures and bounded functions
- Extensions to unbounded functions and to o-finite measures

- The Daniell integral
- The Daniell Integral
- Monotone class theorems
- Measure
- H ̈older, Minkowski , Lp and Lq
- ∥·∥∞ is the essential sup norm
- The Radon-Nikodym Theorem
- The dual space of L1
- The variations of a bounded functional
- Duality of L' and Lq when p(S)

- Integration on locally compact Hausdorff spaces
- Riesz representation theorems
- Fubini’s theorem

- The Riesz representation theorem redux
- Statement of the theorem
- Propositions in topology
- Proof of the uniqueness of the p restricted to B

- Existence
- Definition
- Measurability of the Borel sets
- Compact sets have finite measure
- Interior regularity
- Conclusion of the proof

- Wiener Measure, Brownian motion and white noise
- Wiener measure
- The Big Path Space
- The heat equation
- Paths are continuous with probability one
- Embedding in S’

- Stochastic processes and generalized stochastic processes
- Gaussian measures
- Generalities about expectation and variance
- Gaussian measures and their variances
- The variance of a Gaussian with density
- The variance of Brownian motion

- The derivative of Brownian motion is white noise

- Wiener measure
- Haar measure
- Examples
- R
- Discrete groups
- Lie groups

- Topological facts
- Construction of the Haar integral
- Uniqueness
- p(G) < co if and only if G is compact
- The group algebra
- The involution
- The modular function
- Definition of the involution
- Relation to convolution
- Banach algebras with involutions

- The algebra of finite measures
- Algebras and coalgebras

- Invariant and relatively invariant measures on homogeneous spaces

- Examples
- Branch algebras and the spectral theorem
- Maximal ideals
- Existence
- The maximal spectrum of a ring
- Maximal ideals in a commutative algebra
- Maximal ideals in the ring of continuous functions

- Normed algebras
- The Gelfand representation
- Invertible elements in a Banach algebra form an open set.
- The Gelfand representation for commutative Banach algebras
- The spectral radius
- The generalized Wiener theorem

- Self-adjoint algebras
- An important generalization
- An important application

- The Spectral Theorem for Bounded Normal Operators, Functional Calculus Form
- Statement of the theorem
- SpecB(T) = SpecA(T)
- A direct proof of the spectral theorem.

- Maximal ideals
- The spectral theorem
- Resolutions of the identity
- The spectral theorem for bounded normal operators
- Stone’s formula
- Unbounded operators
- Operators and their domains
- The adjoint
- Self-adjoint operators
- The resolvent
- The multiplication operator form of the spectral theorem
- Cyclic vectors
- The general case
- The spectral theorem for unbounded self-adjoint operators, multiplication operator form.
- The functional calculus
- Resolutions of the identity

- The Riesz-Dunford calculus
- Lorch’s proof of the spectral theorem
- Positive operators
- The point spectrum
- Partition into pure types
- Completion of the proof

- Characterizing operators with purely continuous spectrum
- Appendix. The closed graph theorem

- Stone’s theorem
- von Neumann’s Cayley transform
- An elementary example

- Equibounded semi-groups on a Frechet space
- The infinitesimal generator

- The differential equation
- The resolvent
- Examples

- The power series expansion of the exponential
- The Hille Yosida theorem
- Contraction semigroups
- Dissipation and contraction
- A special case: exp(t(B – I)) with |B| <;

- Convergence of semigroups
- The Trotter product formula
- Lie’s formula
- Chernoff’s theorem
- The product formula
- Commutators
- The Kato-Rellich theorem
- Feynman path integrals

- The Feynman-Kac formula
- The free Hamiltonian and the Yukawa potential
- The Yukawa potential and the resolvent
- The time evolution of the free Hamiltonian

- von Neumann’s Cayley transform
- More about the spectral theorem
- Bound states and scattering states
- Schwartzschild’s theorem
- The mean ergodic theorem
- General considerations
- Using the mean ergodic theorem
- The Amrein-Georgescu theorem
- Kato potentialsApplying the Kato-Rellich method
- Using the inequality (12.7)
- Ruelle’s theorem

- Non-negative operators and quadratic forms
- Fractional powers of a non-negative self-adjoint operator.
- Quadratic forms
- Lower semi-continuous functions
- The main theorem about quadratic forms
- Extensions and cores
- The Friedrichs extension

- Dirichlet boundary conditions
- The Sobolev spaces W1 2(Q) and W ‘2(Q)
- Generalizing the domain and the coefficients
- A Sobolev version of Rademacher’s theorem

- Rayleigh-Ritz and its applications
- The discrete spectrum and the essential spectrum
- Characterizing the discrete spectrum
- Characterizing the essential spectrum
- Operators with empty essential spectrum
- A characterization of compact operators
- The variational method
- Variations on the variational formula
- The secular equation

- The Dirichlet problem for bounded domains
- Valence
- Two dimensional examples
- Hickel theory of hydrocarbons

- Davies’s proof of the spectral theorem
- Symbols
- Slowly decreasing functions
- Stokes’ formula in the plane
- Almost holomorphic extensions
- The Heffler-Sj6strand formula
- A formula for the resolvent
- The functional calculus
- Resolvent invariant subspaces
- Cyclic subspaces
- The spectral representation

- Bound states and scattering states
- Scattering theory
- Examples
- Translation – truncation
- Incoming representations
- Scattering residue

- Breit-Wigner
- The representation theorem for strongly contractive semi-groups
- The Sinai representation theorem
- The Stone – von Neumann theorem

- Examples

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Theory of Functions of a Real Variable