Convex functional analysis

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Authors: Andrew J. Kurdila and Michael Zabarankin

Overview of Book
This book evolved over a period of years as the authors taught classes in variational
calculus and applied functional analysis to graduate students in engineering
and mathematics. The book has likewise been influenced by the authors’ research
programs that have relied on the application of functional analytic principles to
problems in variational calculus, mechanics and control theory.
One of the most difficult tasks in preparing to utilize functional, convex, and
set-valued analysis in practical problems in engineering and physics is the intimidating
number of definitions, lemmas, theorems and propositions that constitute
the foundations of functional analysis. It cannot be overemphasized that functional
analysis can be a powerful tool for analyzing practical problems in mechanics and
physics. However, many academicians and researchers spend their lifetime studying
abstract mathematics. It is a demanding field that requires discipline and
devotion. It is a trite analogy that mathematics can be viewed as a pyramid of
knowledge, that builds layer upon layer as more mathematical structure is put in
place. The difficulty lies in the fact that an engineer or scientist typically would
like to start somewhere “above the base” of the pyramid. Engineers and scientists
are not as concerned, generally speaking, with the subtleties of deriving theorems
axiomatically. Rather, they are interested in gaining a working knowledge of the
applicability of the theory to their field of interest.

The content and structure of the book reflects the sometimes conflicting
requirements of researchers or students who have formal training in either engineering
or applied mathematics. Typically, before taking this course, those trained
within an engineering discipline might have a working knowledge of fundamental
topics in mechanics or control theory. Engineering students may be perfectly comfortable with the notion of the stress distribution in an elastic continuum, or the
velocity field in an incompressible flow. The formulation of the equations governing
the static equilibrium of elastic bodies, or the structure of the Navier-Stokes
Equations for incompressible flow, are often familiar to them. This is usually not
the case for first year graduate students trained in applied mathematics. Rather,
these students will have some familiarity with real analysis or functional analysis.
The fundamental theorems of analysis including the Open Mapping Theorem,
the Hahn-Banach Theorem, and the Closed Graph Theorem will constitute the
foundations of their training in many cases.

Coupled with this essential disparity in the training to which graduate students
in these two disciplines are exposed, it is a fact that formulations and solutions
of modern problems in control and mechanics are couched in functional
analytic terms. This trend is pervasive.
Thus, the goal of the present text is admittedly ambitious. This text seeks
to synthesize topics from abstract analysis with enough recent problems in control
theory and mechanics to provide students from both disciplines with a working
knowledge of functional analysis.

Contents
Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi
1 Classical Abstract Spaces in Functional Analysis
1.1 Introduction and Notation . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Topological Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.2.1 Convergence in Topological Spaces . . . . . . . . . . . . . . 13
1.2.2 Continuity of Functions on Topological Spaces . . . . . . . 15
1.2.3 Weak Topology . . . . . . . . . . . . . . . . . . . . . . . . . 17
1.2.4 Compactness of Sets in Topological Spaces . . . . . . . . . 19
1.3 Metric Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
1.3.1 Convergence and Continuity in Metric Spaces . . . . . . . . 21
1.3.2 Closed and Dense Sets in Metric Spaces . . . . . . . . . . . 23
1.3.3 CompleteMetric Spaces . . . . . . . . . . . . . . . . . . . . 23
1.3.4 The Baire CategoryTheorem . . . . . . . . . . . . . . . . . 25
1.3.5 Compactness of Sets inMetric Spaces . . . . . . . . . . . . 27
1.3.6 Equicontinuous Functions on Metric Spaces . . . . . . . . . 30
1.3.7 The Arzela-Ascoli Theorem . . . . . . . . . . . . . . . . . . 33
1.3.8 H¨older’s and Minkowski’s Inequalities . . . . . . . . . . . . 35
1.4 Vector Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
1.5 Normed Vector Spaces . . . . . . . . . . . . . . . . . . . . . . . . . 45
1.5.1 BasicDefinitions . . . . . . . . . . . . . . . . . . . . . . . . 45
1.5.2 Examples of Normed Vector Spaces . . . . . . . . . . . . . 46
1.6 Space of Lebesgue Measurable Functions . . . . . . . . . . . . . . . 52
1.6.1 Introduction toMeasure Theory . . . . . . . . . . . . . . . 52
1.6.2 Lebesgue Integral . . . . . . . . . . . . . . . . . . . . . . . . 54
1.6.3 Measurable Functions . . . . . . . . . . . . . . . . . . . . . 57
1.7 Hilbert Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
2 Linear Functionals and Linear Operators
2.1 Fundamental Theorems of Analysis . . . . . . . . . . . . . . . . . . 65
2.1.1 Hahn-Banach Theorem . . . . . . . . . . . . . . . . . . . . 65
2.1.2 Uniform Boundedness Theorem . . . . . . . . . . . . . . . . 69
2.1.3 The OpenMapping Theorem . . . . . . . . . . . . . . . . . 71
2.2 Dual Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
2.3 TheWeak Topology . . . . . . . . . . . . . . . . . . . . . . . . . . 79
2.4 The Weak∗ Topology . . . . . . . . . . . . . . . . . . . . . . . . . . 80
2.5 SignedMeasures and Topology . . . . . . . . . . . . . . . . . . . . 88
2.6 Riesz’s Representation Theorem. . . . . . . . . . . . . . . . . . . . 91
2.6.1 Space of Lebesgue Measurable Functions . . . . . . . . . . . 91
2.6.2 Hilbert Spaces . . . . . . . . . . . . . . . . . . . . . . . . . 94
2.7 Closed Operators on Hilbert Spaces . . . . . . . . . . . . . . . . . 95
2.8 Adjoint Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
2.9 Gelfand Triples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
2.10 Bilinear Mappings . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
3 Common Function Spaces in Applications
3.1 The Lp Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
3.2 Sobolev Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
3.2.1 Distributional Derivatives . . . . . . . . . . . . . . . . . . . 114
3.2.2 Sobolev Spaces, Integer Order . . . . . . . . . . . . . . . . . 117
3.2.3 Sobolev Spaces, Fractional Order . . . . . . . . . . . . . . . 118
3.2.4 Trace Theorems . . . . . . . . . . . . . . . . . . . . . . . . 122
3.2.5 The Poincar磂 Inequality . . . . . . . . . . . . . . . . . . . . 123
3.3 Banach Space Valued Functions . . . . . . . . . . . . . . . . . . . . 126
3.3.1 Bochner Integrals . . . . . . . . . . . . . . . . . . . . . . . . 126
3.3.2 The Space Lp
(0, T ),X . . . . . . . . . . . . . . . . . . . . 131
3.3.3 The Space Wp,q
(0, T ),X . . . . . . . . . . . . . . . . . . 133
4 Differential Calculus in Normed Vector Spaces
4.1 Differentiability of Functionals . . . . . . . . . . . . . . . . . . . . 137
4.1.1 Gateaux Differentiability . . . . . . . . . . . . . . . . . . . 137
4.1.2 Fr磂chet Differentiability . . . . . . . . . . . . . . . . . . . . 139
4.2 Classical Examples of Differentiable Operators . . . . . . . . . . . 143
5 Minimization of Functionals
5.1 TheWeierstrass Theorem . . . . . . . . . . . . . . . . . . . . . . . 161
5.2 Elementary Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . 163
5.3 Minimization of Differentiable Functionals . . . . . . . . . . . . . . 165
5.4 Equality Constrained Smooth Functionals . . . . . . . . . . . . . . 166
5.5 Fr磂chetDifferentiable Implicit Functionals . . . . . . . . . . . . . . 171
6 Convex Functionals
6.1 Characterization of Convexity . . . . . . . . . . . . . . . . . . . . . 177
6.2 Gateaux Differentiable Convex Functionals . . . . . . . . . . . . . 180
6.3 Convex Programming in Rn . . . . . . . . . . . . . . . . . . . . . . 183
6.4 OrderedVector Spaces . . . . . . . . . . . . . . . . . . . . . . . . . 188
6.4.1 Positive Cones, Negative Cones, and Orderings . . . . . . . 189
6.4.2 Orderings on Sobolev Spaces . . . . . . . . . . . . . . . . . 191
6.5 Convex Programming in Ordered Vector Spaces . . . . . . . . . . . 193
6.6 Gateaux Differentiable Functionals on Ordered Vector Spaces . . . 199
7 Lower Semicontinuous Functionals
7.1 Characterization of Lower Semicontinuity . . . . . . . . . . . . . . 205
7.2 Lower Semicontinuous Functionals and Convexity . . . . . . . . . . 208
7.2.1 Banach Theorem for Lower Semicontinuous Functionals . . 208
7.2.2 Gateaux Differentiability . . . . . . . . . . . . . . . . . . . 210
7.2.3 Lower Semicontinuity in Weak Topologies . . . . . . . . . . 210
7.3 The GeneralizedWeierstrass Theorem . . . . . . . . . . . . . . . . 212
7.3.1 Compactness inWeak Topologies . . . . . . . . . . . . . . . 213
7.3.2 Bounded Constraint Sets . . . . . . . . . . . . . . . . . . . 215
7.3.3 Unbounded Constraint Sets . . . . . . . . . . . . . . . . . . 215
7.3.4 Constraint Sets on Ordered Vector Spaces . . . . . . . . . . 217
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223

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Convex functional analysis: 4,5,6,7章

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Convex functional analysis: 参考文献