Theory of functions of a real variable.

1 Theory of functions of a real SternbergMay 10, have taught the beginning graduate course in real variables and functionalanalysis three times in the last five years, and this book is the result. Thecourse assumes that the student has seen the basics of real variable Theory andpoint set topology. The elements of the topology of metrics spaces are presented(in the nature of a rapid review) in Chapter course itself consists of two parts: 1) measure Theory and integration,and 2) Hilbert space Theory , especially the spectral theorem and its Chapter II I do the basics of Hilbert space Theory , what I can dowithout measure Theory or the Lebesgue integral. The hero here (and perhapsfor the first half of the course) is the Riesz representation theorem. Includedis the spectral theorem for compact self-adjoint operators and applications ofthis theorem to elliptic partial differential equations.

2 The pde material followsclosely the treatment by Bers and Schecter inPartial Differential EquationsbyBers, John and Schecter AMS (1964)Chapter III is a rapid presentation of the basics about the Fourier IV is concerned with measure Theory . The first part follows Caratheodory sclassical presentation. The second part dealing with Hausdorff measure and di-mension, Hutchinson s theorem and fractals is taken in large part from the bookby Edgar,Measure Theory , Topology, and Fractal GeometrySpringer (1991).This book contains many more details and beautiful examples and V is a standard treatment of the Lebesgue VI, and VIII deal with abstract measure Theory and chapters basically follow the treatment by Loomis in hisAbstract Har-monic VII develops the Theory of Wiener measure and Brownian motionfollowing a classical paper by Ed Nelson published in the Journal of Mathemat-ical Physics in 1964.

3 Then we study the idea of a generalized random processas introduced by Gelfand and Vilenkin, but from a point of view taught to usby Dan rest of the book is devoted to the spectral theorem. We present threeproofs of this theorem. The first, which is currently the most popular, derivesthe theorem from the Gelfand representation theorem for Banach algebras. Thisis presented in Chapter IX (for bounded operators). In this chapter we againfollow Loomis rather Chapter X we extend the proof to unbounded operators, following Loomisand Reed and SimonMethods of Modern Mathematical Physics. Then we giveLorch s proof of the spectral theorem from his bookSpectral Theory . This hasthe flavor of complex analysis. The third proof due to Davies, presented at theend of Chapter XII replaces complex analysis by almost complex remaining chapters can be considered as giving more specialized in-formation about the spectral theorem and its applications.

4 Chapter XI is de-voted to one parameter semi-groups, and especially to Stone s theorem aboutthe infinitesimal generator of one parameter groups of unitary XII discusses some theorems which are of importance in applications of3the spectral theorem to quantum mechanics and quantum chemistry. ChapterXIII is a brief introduction to the Lax-Phillips Theory of The topology of metric 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.

5 Machado s theorem.. The Hahn-Banach theorem.. The Uniform Boundedness Principle.. 352 Hilbert Spaces and Compact 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 isL2(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 operatorddx.

6 G arding s inequality, special case.. The Heisenberg uncertainty principle.. The Sobolev Spaces.. G arding s inequality.. Consequences of G arding s inequality.. Extension of the basic lemmas to manifolds.. Example: Hodge Theory .. The resolvent.. 833 The Fourier Conventions, especially about 2 .. 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 ofS .. 934 Measure Lebesgue outer measure.. Lebesgue inner measure.. Lebesgue s definition of measurability.

7 Caratheodory s definition of measurability.. Countable additivity.. -fields, measures, and outer measures.. Constructing outer measures, Method I.. A pathological example.. Metric outer measures.. Constructing outer measures, Method II.. An example.. 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 .. 1295 The Lebesgue Real valued measurable functions .. The integral of a non-negative function.. Fatou s lemma.. The monotone convergence theorem.. The spaceL1(X,R).. The dominated convergence theorem.. Riemann integrability.. The Beppo - Levi theorem.

8 Complete.. Dense subsets ofL1(R,R).. The Riemann-Lebesgue Lemma.. The Cantor-Lebesgue theorem.. Fubini s theorem.. Product -fields.. -systems and -systems.. The monotone class theorem.. Fubini for finite measures and bounded functions .. Extensions to unbounded functions and to -finite The Daniell The Daniell Integral .. Monotone class theorems.. Measure.. H older, Minkowski ,LpandLq.. is the essential sup norm.. The Radon-Nikodym Theorem.. The dual space ofLp.. The variations of a bounded functional.. Duality ofLpandLqwhen (S)< .. The case where (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.

9 Proof of the uniqueness of the restricted toB(X).. Existence.. Definition.. Measurability of the Borel sets.. Compact sets have finite measure.. Interior regularity.. Conclusion of the proof.. 1847 Wiener measure, Brownian motion and white Wiener measure.. The Big Path Space.. The heat equation.. Paths are continuous with probability one.. Embedding inS .. 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.. 2028 Haar Examples.. Discrete groups.. Lie groups.. Topological facts.. Construction of the Haar integral.

10 Uniqueness.. (G)< if and only ifGis 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 Banach algebras and the spectral 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 al-gebras.. The spectral radius.. The generalized Wiener theorem.. Self-adjoint algebras.