"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 are introduced 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 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.
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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.