Applied Functional Analysis (2^nd Ed., 2nd ed. 1981) a Stochastic Modelling and Applied Probability Series, Vol. 3

In preparing the second edition, I have taken advantage of the opportunity to correct errors as well as revise the presentation in many places. New material has been included, in addition, reflecting relevant recent work. The help of many colleagues (and especially Professor J. Stoer) in ferreting out errors is gratefully acknowledged. I also owe special thanks to Professor v. Sazonov for many discussions on the white noise theory in Chapter 6. February, 1981 A. V. BALAKRISHNAN v Preface to the First Edition The title "Applied Functional Analysis" is intended to be short for "Functional analysis in a Hilbert space and certain of its applications," the applications being drawn mostly from areas variously referred to as system optimization or control systems or systems analysis. One of the signs of the times is a discernible tilt toward application in mathematics and conversely a greater level of mathematical sophistication in the application areas such as economics or system science, both spurred undoubtedly by the heightening pace of digital computer usage. This book is an entry into this twilight zone. The aspects of functional analysis treated here are rapidly becoming essential in the training at the advance graduate level of system scientists and/or mathematical economists. There are of course now available many excellent treatises on functional analysis.

1 Basic Properties of Hilbert Spaces.- 1.0 Introduction.- 1.1 Basic Definitions.- 1.2 Examples of Hilbert Spaces.- 1.3 Hilbert Spaces from Hilbert Spaces.- 1.4 Convex Sets and Projections.- 1.5 Orthogonality and Orthonormal Bases.- 1.6 Continuous Linear Functionals.- 1.7 Riesz Representation Theorem.- 1.8 Weak Convergence.- 1.9 Nonlinear Functionals and Generalized Curves.- 1.10 The Hahn-Banach Theorem.- 2 Convex Sets and Convex Programming.- 2.0 Introduction.- 2.1 Elementary Notions.- 2.2 Support Functional of a Convex Set.- 2.3 Minkowski Functional.- 2.4 The Support Mapping.- 2.5 Separation Theorem.- 2.6 Application to Convex Programming.- 2.7 Generalization to Infinite Dimensional Inequalities.- 2.8 A Fundamental Result of Game Theory: Minimax Theorem.- 2.9 Application: Theorem of Farkas.- 3 Functions, Transformations, Operators.- 3.0 Introduction.- 3.1 Linear Operators and their Adjoints.- 3.2 Spectral Theory of Operators.- 3.3 Spectral Theory of Compact Operators.- 3.4 Operators on Separable Hilbert Spaces.- 3.5 L2 Spaces over Hilbert Spaces.- 3.6 Multilinear Forms.- 3.7 Nonlinear Volterra Operators.- 4 Semigroups of Linear Operators.- 4.0 Introduction.- 4.1 Definitions and General Properties of Semigroups.- 4.2 Generation of Semigroups.- 4.3 Semigroups over Hilbert Spaces: Dissipative Semigroups.- 4.4 Compact Semigroups.- 4.5 Analytic (Holomorphic) Semigroups.- 4.6 Elementary Examples of Semigroups.- 4.7 Extensions.- 4.8 Differential Equations: Cauchy Problem.- 4.9 Controllability.- 4.10 State Reduction: Observability.- 4.11 Stability and Stabilizability.- 4.12 Boundary Input: An Example.- 4.13 Evolution Equations.- 5 Optimal Control Theory.- 5.0 Introduction.- 5.1 Preliminaries.- 5.2 Linear Quadratic Regulator Problem.- 5.3 Linear Quadratic Regulator Problem: Infinite Time Interval.- 5.4 Hard Constraints.- 5.5 Final Value Control.- 5.6 Time Optimal Control Problem.- 6 Stochastic Optimization Theory.- 6.0 Introduction.- 6.1 Preliminaries.- 6.2 Measures on Cylinder Sets.- 6.3 Characteristic Functions and Countable Additivity.- 6.4 Weak Random Variables.- 6.5 Random Variables.- 6.6 White Noise.- 6.7 Differential Systems.- 6.8 The Filtering Problem.- 6.9 Stochastic Control.- 6.10 Physical Random Variables.- 6.11 Radon-Nikodym Derivatives.- 6.12 Nonlinear Stochastic Equations.