تحلیل واقعی و انتزاعی

I Topology, Continuity, Algebra, Derivatives Some Basic Topics Basic Definitions The Schroder Bernstein Theorem Equivalence Relations sup and inf Double Series lim sup and lim inf Nested Interval Lemma The Hausdorff Maximal Theorem Metric Spaces Open and Closed Sets, Sequences, Limit Points Cauchy Sequences, Completeness Closure of a Set Separable Metric Spaces Compact Sets Continuous Functions Continuity and Compactness Lipschitz Continuity and Contraction Maps Convergence of Functions Compactness in C( X,Y) Ascoli Arzela Theorem Connected Sets Partitions of Unity in Metric Space Exercises Linear Spaces Algebra in Fn, Vector Spaces Subspaces Spans and Bases Inner Product and Normed Linear Spaces The Inner Product in Fn General Inner Product Spaces Normed Vector Spaces The p Norms Orthonormal Bases Equivalence of Norms Covering Theorems Vitali Covering Theorem Besicovitch Covering Theorem Exercises Functions on Normed Linear Spaces L( V,W) as a Vector Space The Norm of a Linear Map, Operator Norm Continuous Functions in Normed Linear Space Polynomials Weierstrass Approximation Theorem Functions of Many Variables A Generalization with Tietze Extension Theorem An Approach to the Integral The Stone Weierstrass Approximation Theorem Connectedness in Normed Linear Space Saddle Points Exercises Fixed Point Theorems Simplices and Triangulations Labeling Vertices The Brouwer Fixed Point Theorem The Schauder Fixed Point Theorem The Kakutani Fixed Point Theorem Ekeland's Variational Principle Cariste Fixed Point Theorem A Density Result Exercises The Derivative Limits of a Function Basic Definitions The Chain Rule The Matrix of the Derivative A Mean Value Inequality Existence of the Derivative, C1 Functions Higher Order Derivatives Some Standard Notation The Derivative and the Cartesian Product Mixed Partial Derivatives A Cofactor Identity Newton's Method Exercises Implicit Function Theorem Statement and Proof of the Theorem More Derivatives The Case of Rn Exercises The Method of Lagrange Multipliers The Taylor Formula Second Derivative Test The Rank Theorem The Local Structure of C1 Mappings Invariance of Domain Exercises II Integration Abstract Measures and Measurable Functions Simple Functions and Measurable Functions Measures and their Properties Dynkin's Lemma Measures and Regularity Outer Measures Exercises An Outer Measure on P( R) Measures From Outer Measures When do the Measurable Sets Include Borel Sets? One Dimensional Lebesgue Stieltjes Measure Completion of a Measure Space Vitali Coverings Differentiation of Increasing Functions Exercises Multifunctions and Their Measurability The General Case A Special Case When ( 0=x"0121) Compact Kuratowski's Theorem Measurability of Fixed Points Other Measurability Considerations Exercises The Abstract Lebesgue Integral Definition for Nonnegative Measurable Functions Riemann Integrals for Decreasing Functions The Lebesgue Integral for Nonnegative Functions Nonnegative Simple Functions The Monotone Convergence Theorem Other Definitions Fatou's Lemma The Integral's Righteous Algebraic Desires The Lebesgue Integral, L1 The Dominated Convergence Theorem Some Important General Theory Eggoroff's Theorem The Vitali Convergence Theorem One Dimensional Lebesgue Stieltjes Integral The Distribution Function Good Lambda Inequality Radon Nikodym Theorem Abstract Product Measures and Integrals Exercises Regular Measures Regular Measures in a Metric Space Differentiation of Radon Measures Maximal Functions, Fundamental Theorem of Calculus Symmetric Derivative for Radon Measures Radon Nikodym Theorem for Radon Measures Absolutely Continuous Functions Constructing Measures from Functionals The p Dimensional Lebesgue Measure The Brouwer Fixed Point Theorem Exercises Change of Variables, Linear Maps Differentiable Functions and Measurability Change of Variables, Nonlinear Maps Mappings which are not One to One Spherical Coordinates Exercises Integration on Manifolds Relatively Open Sets Manifolds The Area Measure on a Manifold Exercises Divergence Theorem Volumes of Balls in Rp Exercises The Lp Spaces Basic Inequalities and Properties Density Considerations Separability Continuity of Translation Mollifiers and Density of Smooth Functions Smooth Partitions of Unity Exercises Degree Theory Sard's Lemma and Approximation Properties of the Degree Borsuk's Theorem Applications Product Formula, Jordan Separation Theorem General Jordan Separation Theorem Uniqueness of the Degree Exercises Hausdorff Measure Lipschitz Functions Lipschitz Functions and Gateaux Derivatives Rademacher's Theorem Weak Derivatives Definition of Hausdorff Measures Properties of Hausdorff Measure Hp and mp Technical Considerations Steiner Symmetrization The Isodiametric Inequality The Proper Value of 0=x"010C( p) A Formula for 0=x"010B( p) The Area Formula Estimates for Hausdorff Measure Comparison Theorems A Decomposition Estimates and a Limit The Area Formula Mappings that are Not One to One The Divergence Theorem The Reynolds Transport Formula The Coarea Formula Change of Variables Orientation in Higher Dimensions The Wedge Product The Exterior Derivative Stoke's Theorem Green's Theorem and Stokes Theorem The Divergence Theorem Exercises III Abstract Theory Hausdorff Spaces and Measures General Topological Spaces The Alexander Sub-basis Theorem Stone Weierstrass Theorem The Case of Locally Compact Sets The Case of Complex Valued Functions Partitions of Unity Measures on Hausdorff Spaces Measures and Positive Linear Functionals Slicing Measures Exercises Product Measures Measure on Infinite Products Algebras Caratheodory Extension Theorem Kolmogorov Extension Theorem Exercises Banach Spaces Theorems Based on Baire Category Baire Category Theorem Uniform Boundedness Theorem Open Mapping Theorem Closed Graph Theorem Hahn Banach Theorem Partially Ordered Sets Gauge Functions and Hahn Banach Theorem The Complex Version of the Hahn Banach Theorem The Dual Space and Adjoint Operators Uniform Convexity of Lp Closed Subspaces Weak And Weak Topologies Basic Definitions Banach Alaoglu Theorem Eberlein Smulian Theorem Differential Equations Exercises Hilbert Spaces Basic Theory The Hilbert Space L( U) Approximations in Hilbert Space Orthonormal Sets Compact Operators Compact Operators in Hilbert Space Nuclear Operators Hilbert Schmidt Operators Square Roots Ordinary Differential Equations in Banach Space Fractional Powers of Operators General Theory of Continuous Semigroups An Evolution Equation Adjoints, Hilbert Space Adjoints, Reflexive Banach Space Exercises Representation Theorems Radon Nikodym Theorem Vector Measures Representation for the Dual Space of Lp Weak Compactness The Dual Space of L( ) Non 0=x"011B Finite Case The Dual Space of C0( X) Extending Righteous Functionals The Riesz Representation Theorem Exercises Fourier Transforms Fourier Transforms of Functions In G Fourier Transforms of just about Anything Fourier Transforms of G Fourier Transforms of Functions in L1( Rn) Fourier Transforms Of Functions In L2( Rn) The Schwartz Class Convolution Exercises The Bochner Integral Strong and Weak Measurability Eggoroff's Theorem The Bochner Integral Definition and Basic Properties Taking a Closed Operator Out of the Integral Operator Valued Functions Review of Hilbert Schmidt Theorem Measurable Compact Operators Fubini's Theorem for Bochner Integrals The Spaces Lp( ;X) Measurable Representatives Vector Measures The Riesz Representation Theorem An Example of Polish Space Pointwise Behavior of Weakly Convergent Sequences Some Embedding Theorems Conditional Expectation in Banach Spaces Exercises Stone's Theorem and Partitions of Unity Partitions of Unity and Stone's Theorem An Extension Theorem, Retracts IV Stochastic Processes and Probability Independence Random Variables and Independence Kolmogorov Extension Theorem Independent Events and 0=x"011B Algebras Independence for Banach Space Valued Random Variables Reduction to Finite Dimensions 0,1 Laws Kolmogorov's Inequality, Strong Law of Large Numbers Analytical Considerations The Characteristic Function Conditional Probability Conditional Expectation, Sub-martingales Characteristic Functions and Independence Characteristic Functions for Measures Independence in Banach Space Convolution and Sums The Normal Distribution The Multivariate Normal Distribution Linear Combinations Finding Moments Prokhorov and Levy Theorems The Central Limit Theorem Conditional Expectation and Martingales Conditional Expectation Conditional Expectation and Independence Discrete Stochastic Processes Upcrossings The Sub-martingale Convergence Theorem Doob Sub-martingale Estimates Optional Sampling and Stopping Times Optional Sampling for Martingales Optional Sampling Theorem for Sub-Martingales Reverse Sub-martingale Convergence Theorem Strong Law of Large Numbers Continuous Stochastic Processes Fundamental Definitions and Properties Kolmogorov Centsov Continuity Theorem Filtrations Martingales Some Maximal Estimates Optional Sampling Theorems Review of Discreet Stopping Times Review of Doob Optional Sampling Theorem Doob Optional Sampling Continuous Case Stopping Times The Optional Sampling Theorem Continuous Case Maximal Inequalities and Stopping Times Continuous Sub-martingale Convergence Hitting This Before That Review of Some Linear Algebra The Matrix of a Linear Map Block Multiplication of Matrices Schur's Theorem Hermitian and Symmetric Matrices The Right Polar Factorization Elementary matrices The Row Reduced Echelon Form Of A Matrix Finding the Inverse of a Matrix The Mathematical Theory of Determinants The Function sgn The Definition of the Determinant A Symmetric Definition Basic Properties of the Determinant Expansion Using Cofactors A Formula for the Inverse Cramer's Rule Rank of a Matrix An Identity of Cauchy The Cayley Hamilton Theorem