Intermediate Mathematical Analysis

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Intermediate Mathematical Analysis aims at presenting advanced topics such as continuity, uniform continuity, tests of convergence of series, uniform convergence of series, power series, polynomial approximations and Fourier series in a more general setting. Metric and Normed Linear Spaces areintroduced at an early stage and are used wherever found advantageous. The book places a consistent emphasis on showing the power of the classical analysis by applying it to the study of real valued functions and their applications. Requiring only a nodding acquaintance with ? – d type arguments and definition of Riemann integral, the book will provide sufficient background material for studies in Functional Analysis, Topology, Complex Analysis, Theory of Differential Equations and so on. Key Features Theorems are proved in complete details and illustrated profusely using examples and counter examples. About hundred illustrations and four hundred exercises, most of which have hints or solutions at the end of the book. End of the chapter projects to challenge ambitious readers. Table of Contents Preface  Metric Spaces: Definition of a Metric Space  Some Examples  Euclidean Metric in Kn  Normed Spaces: Normed Spaces  The Sequence Space lp, p>1:  Topology of Metric Spaces: Topological Spaces  Topology of Metric Spaces  Equivalent Metrics  Subspaces  Sequences in Metric Spaces  Closure, Interior and Boundary  Complete Metric Spaces: The Cauchy Sequence  Subsequences  Contraction Mapping  Continuity: Continuity between Metric Spaces  Open Maps, Closed Maps  Uniform Continuity  Homeomorphism and Isometry  Discontinuities and All That  Connected Metric Spaces: Connected Metric Spaces  Application of Intermediate Value Theorem  Connected Components  Path Connected Spaces  Compact Metric Spaces: Compactness  Characterization of Compact Metric Spaces  Applications  Sequences and Series of Functions: Pointwise Convergence  Uniform Convergence  Power Series: Limit Superior and Limit Inferior  Power Series  The Circular Functions  The Exponential Function  Fourier Series: Orthogonal Functions  Fourier Sine and Cosine Series  Mean Square Convergence of Fourier Series  The Pointwise Convergence of Fourier Series  Appendixes  Bibliography  Index  Index of Symbols.