1/General Topology. Topological Spaces.- 1.0 Introduction.- 1.1 Sets. Functions.- 1.2 Topology and Topological Spaces.- 1.3 Compactness in Topological Spaces.- 1.4 Metric Spaces. Examples and Some Properties.- 1.5 Measures of Noncompactness in Metric Spaces.- 1.6 Some Historical Remarks.- 2/Banach Spaces and Complete Inner Product Spaces.- 2.0 Introduction.- 2.1 Linear Spaces. Sets in Linear Spaces.- 2.2 Normed Linear Spaces and Banach Spaces.- 2.3 The Extension Theorems.- 2.4 Linear Operators on Banach Spaces. Classes of Linear Operators.- 2.5 Three Basic Theorems of Linear Functional Analysis.- 2.6 Inner Product Spaces. Definitions and Some Examples.- 2.7 Von Neumann Generalized Direct Sums.- 2.8 Tensor Products of Banach Spaces and of Complete Inner Product Spaces.- 3/Orthogonality and Bases.- 3.0 Introduction.- 3.1 Orthogonality in Linear Spaces with an Inner Product.- 3.2 Bases in Complete Inner Product Spaces.- 3.3 Subspaces in Spaces with an Inner Product. The Orthogonal Decomposition.- 3.4 Some Applications of the Fréchet-Riesz Representation Theorem.- 3.5 Some Examples of Bases in Concrete Complete Inner Product Spaces.- 3.6 Perturbation of Bases in Complete Inner Product Spaces.- 3.7 Some Classes of Bases (Hardy Bases) and the Theory of Communication.- 4/Metric Characterizations of Inner Product Spaces.- 4.0 Introduction.- 4.1 Inner Product Structures on Linear Spaces.- 4.2 Inner Product Structures and Complexification.- 4.3 The Fréchet and Jordan-von Neumann Characterization of Inner Product Spaces.- 4.4 The Ficken Characterization of Inner Product Spaces.- 4.5 Closed Maximal Linear Subspaces and Inner Product Structures.- 4.6 Loewner's Ellipses. Ellipsoids.- 4.7 Ellipses and Inner Product Spaces.- 4.8 The Integral Form of the Parallelogram Law.- 4.9 Topological Inner Productability.- 4.10 Local Norm Characterizations of Inner Product Structures.- 4.11 Other Norm Characterizations of Inner Product Structures.- 4.12 Orthogonality in Normed Linear Spaces and Characterizations of Inner Product Spaces.- 4.13 Approximation Theory and Characterizations of Inner Product Spaces.- 4.14 Chebyshev Centers and Inner Product Structures.- 4.15 On Some Norms on Two-Dimensional Spaces.- 4.16 Parameters Associated with Normed Linear Spaces and Inner Product Structures.- 4.17 The Modulus of Convexity and the Modulus of Smoothness and Inner Product Spaces.- 4.18 Spaces Isomorphic to Inner Product Spaces.- 4.19 Inner Product Spaces and Classes of Metric Spaces.- 4.20 Other Metric Characterizations of Inner Product Spaces.- 4.21 Angles and Complete Inner Product Spaces.- 5/Banach Algebras.- 5.0 Introduction.- 5.1 Definition of Banach Algebras and Some Examples.- 5.2 Ideals in Banach Algebras.- 5.3 The Spectrum of an Element in a Complex Banach Algebra with Identity.- 5.4 The Gelfand Representation. The Representation and Structure of Commutative Banach Algebras.- 5.5 The Representations of B*-Algebras with Identity.- 5.6 Approximate Identities in Banach Algebras.- 5.7 Classes of Elements in Banach Algebras.- 6/Bounded and Unbounded Linear Operators.- 6.0 Introduction.- 6.1 Classes of Bounded Linear Operators on Complete Inner Product Spaces.- 6.2 Normal, Unitary and Partial Isometry Operators.- 6.3 Semispectral and Spectral Families of Radon Measures.- 6.4 Unbounded Operators.- 6.5 Closed and Closable Operators.- 6.6 The Graph of Linear Operators and Some Applications.- 6.7 Hermitian, Selfadjoint and Essentially Selfadjoint Operators.- 6.8 Some Examples of Selfadjoint and Essentially Selfadjoint Operators.- 6.9 Selfadjoint Extensions.- 6.10 Extensions of Semibounded Linear Operators.- 6.11 Unbounded Normal Operators and Some Related Classes of Operators.- 6.12 Some Decomposition Theorems.- 7/Ideals of Operators on Complete Inner Product Spaces and on Banach Spaces.- 7.0 Introduction.- 7.1 Some Terminology and Notations.- 7.2 Ideals of Operators on Complete Inner Product Spaces.- 7.3 The Banach Spaces Cp.- 7.4 Ideal Sets and Ideals of Compact Operators.- 7.5 Banach Ideals. Classes of Summing Operators.- 7.6 Grothendieck's Fundamental Theorem.- 7.7 On the Coincidence of Classes of Absolutely p-Summing Operators.- 7.8 Types, Cotypes and Rademacher Averages in Banach Spaces.- 8/Operator Characterizations of Inner Product Spaces.- 8.0 Introduction.- 8.1 O-Negative Definite Functions and Inner Product Spaces.- 8.2 Some Inequalities and a Characterization of Inner Product Spaces.- 8.3 Nonexpansive Mappings and the Extension Problem.- 8.4 Fixed Point Sets for Nonexpansive Mappings and Inner Product Structures.- 8.5 Support Mappings and Inner Product Structures.- 8.6 Smooth Functions on Banach Spaces and Inner Product Structures.- 8.7 Classes of Functions on Banach Spaces and Inner Product Structures.- 8.8 Linear Operators and Inner Product Structures.- 8.9 Algebraic Characterizations of Inner Product Structures.- 8.10 Hermitian Decomposition of a Banach Space and Inner Product Spaces.- 8.11 Classes of Hermitian Elements and Inner Product Structures.- 8.12 A Variational Characterization of Inner Product Structures.- 8.13 Von Neumann Spectral Sets and a Characterization of Inner Product Spaces.- 8.14 A Series-Immersed Isomorphic Characterization of Complete Inner Product Spaces.- 8.15 A Symmetric-Invariant Characterization of L2[0,1]..- 9/Probability Theory and Inner Product Structures.- 9.0 Introduction.- 9.1 Probabilities on Banach Spaces.- 9.2 Bernoulli and Gaussian Random Independent Variables and Inner Product Structures.- 9.3 Biconvex Functions and a Characterization of Complete Inner Product Spaces.- 9.4 Other Probabilistic Characterizations of Inner Product Structures.- 10/Positive Definite Functions, Functions of Positive Type and Inner Product Structures.- 10.0 Introduction.- 10.1 Positive Definite Functions. Definitions and Some Examples.- 10.2 The Coincidence of Classes of Positive Definite Functions and Functions of Positive Type on Locally Compact Abelian Groups.- 10.3 Completely Positive Maps. Stinespring's Theorem.- 10.4 The Nevanlinna Problem.- 10.5 The Monotone Functions of C. Loewner.- 11/Reproducing Kernels and Inner Product Spaces. Applications.- 11.0 Introduction.- 11.1 Reproducing Kernels. Basic Properties.- 11.2 Linear Functionals and Linear Operators on Spaces with Reproducing Kernels.- 11.3 Some Properties of Reproducing Kernels.- 11.4 Functional Completion of a Space, of Functions. The Existence of Complete Inner Product Spaces with Reproducing Kernels.- 11.5 Some Examples of Complete Inner Product Spaces with Reproducing Kernels.- 11.6 Operations on the Reproducing Kernel Functions (the Sum, Products and Limits of Reproducing Kernels).- 11.7 Interpolation, Extremal and Minimal Problems and Reproducing Kernels.- 11.8 Conformal Mappings and Reproducing Kernel Functions.- 11.9 Invariant Subspaces for Generalized Translations and Reproducing Kernels.- 11.10 Spline Functions and Inner Product Spaces with Reproducing Kernels.- 11.11 Dilation Theory and Reproducing Kernels.- 11.12 Some Applications of Reproducing Kernels.- 12/Inner Product Modules.- 12.0 Introduction.- 12.1 Inner Product Modules. Definition and Some Examples. Bounded Module Maps.- 12.2 Some Representation Theorems.- 12.3 Dilation Theory and Inner Product Modules.- 12.4 Von Neumann Algebra Module.- 12.5 The AW*-Modules of Kaplansky.- 12.6 Classes of Kaplansky's Inner Product Modules.- 13/Quaternionic Complete Inner Product Spaces.- 13.0 Introduction.- 13.1 The Quaternions.- 13.2 Linear Spaces over Quaternions.- 13.3 The Symplectic Image of a Left Quaternionic Complete Inner Product Space.- 13.4 Classes of Operators on Left Quaternionic Inner Product Spaces.- 13.5 Spectral Theory on Left Quaternionic Complete Inner Product Spaces.- 13.6 Functional Calculus for Operators on Left Quaternionic Complete Inner Product Spaces.- 14/Inner Product Algebras.- 14.0 Introduction.- 14.1 Inner Product Algebras.- 14.2 Complete Inner Product Algebras with Identity.- 14.3 H*-Algebras.- 14.4 Inner Product Algebras and H*-Algebras.- 15/Non-Archimedean, Nonstandard, Intuitionistic and Constructive Inner Product Spaces.- 15.0 Introduction.- 15.1 Non-Archimedean Normed Linear Spaces. Non-Archimedean Inner Product Spaces.- 15.2 Nonstandard Inner Product Spaces.- 15.3 Intuitionistic Complete Inner Product Spaces.- 15.4 Constructive Inner Product Spaces.- 16/Indefinite Inner Product Structures.- 16.0 Introduction.- 16.1 Indefinite Inner Product Linear Spaces.- 16.2 Orthogonality and Orthogonal Decomposition.- 16.3 Linear Operators on Spaces with an Indefinite Inner Product.- 16.4 Some Classes of Spaces with an Indefinite Metric.- 16.5 Modules with an Indefinite Inner Product.- 17/Some Applications of Inner Product Structures.- 17.0 Introduction.- 17.1 Certain Applications of the Cauchy-Buniakowsky Inequality to Some Extremal Problems.- 17.2 Invariant Subspaces for the Shift.- 17.3 Fourier Transforms and the Plancherel Theorem.- 17.4 The Sturm-Liouville Problem and Inner Product Spaces.- 17.5 Measures of Dependence of Random Variables and Inner Product Spaces.- 17.6 Ergodic Theory and Complete Inner Product Spaces.- 17.7 Classes of Stochastic Processes and Inner Product Structures.- 17.8 Inner Products and Differentials. Harmonic and Analytic Differentials.- 17.9 Differential Geometry in Complete Inner Product Spaces.- 17.10 Univalent Functions and Complete Inner Product Spaces.- 17.11 Complete Inner Product Spaces and Roots of Polynomials (and Analytic Functions).- 17.12 Bohr's Basic Theorem of Almost Periodic Functions.- 17.13 Inner Product Structures of Lie Algebras and Jordan Algebras.- 17.14 Potential Theory and Inner Product Structures.- 17.15 Gravity Theory, Statistical Physics and Dynamics and Inner Product Structures.- 17.16 Quantum Mechanics and Operators on Complete Inner Product Spaces.- 17.17 Number Theory and Complete Inner Product Spaces.- 18/A Collection of Problems.- 18.0 Introduction.- 18.1 Problems on Inner Product Structures.- References.- List of Symbols and Abbreviations.- Author Index.

Approach your problems from the right end It isn't that they can't see the solution. It is and begin with the answers. Then one day, that they can't see the problem. perhaps you will find the final question. G. K. Chesterton. The Scandal of Father 'The Hermit Oad in Crane Feathers' in R. Brown 'The point of a Pin'. van Gulik's The Chinese Maze Murders. Growing specialization and diversification have brought a host of monographs and textbooks on increasingly specialized topics. However, the "tree" of knowledge of mathematics and related fields does not grow only by putting forth new branches. It also happens, quite often in fact, that branches which were thought to be completely disparate are suddenly seen to be related. Further, the kind and level of sophistication of mathematics applied in various sciences has changed drastically in recent years: measure theory is used (non-trivially) in regional and theoretical economics; algebraic geometry interacts with physics; the Minkowsky lemma, coding theory and the structure of water meet one another in packing and covering theory; quantum fields, crystal defects and mathematical programming profit from homotopy theory; Lie algebras are relevant to filtering; and prediction and electrical engineering can use Stein spaces. And in addition to this there are such new emerging subdisciplines as "experimental mathematics", "CFD", "completely integrable systems", "chaos, synergetics and large-scale order", which are almost impossible to fit into the existing classification schemes. They draw upon widely different sections of mathematics.