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This preliminary chapter contains notations, definitions, and basic concepts needed for the study of Measure Theory and Functional Analysis. Most of this chapter is for reference and may be read only as needed. Included are concepts such as Convergence, Continuity, and Compactness in Euclidean Spaces. The theory of Euclidean Measure and sets of Measure Zero are covered. An overview is included of Integration sufficient to begin the study of Functional Analysis. The chapter finishes with topics such as Functions of Bounded Variation, Inequalities, along with a discussion of the Axiom of Choice.
This chapter extends Lebesgue measure to Abstract Measure Spaces. Example such as Lebesgue-Stieltjes Measures, Probability Measures, and Signed Measures are considered. The Radon–Nikodym Theorem on Absolute Continuity is proven. The Radon–Nikodym derivative is defined and its Chain Rule is proven.
Many applications of Functional Analysis are introduced, including Least Squares Approximation Methods, the Vibrating String or Membrane (the Wave Equation), Heat Flow on a rod or plate (the Heat Equation), Gambler's Ruin and Random Walk, Sampling Theorem of Signal Processing, the Atomic Theory of Matter, Uncertainty Principle, and Wavelets. The beautiful connection between Group Theory, Fourier Series, and the Haar Integral (which for Euclidean Space, is the Lebesgue Integral) is investigated.
Inner Product Spaces and their Fourier Series are studied in this chapter, including Hilbert Spaces and Adjoint Operators. Convergence of Trigonometric and Square Wave Fourier Series of Integrable Functions are investigated. Theorems connecting Summability of Fourier Series and Summability Kernels are studied. The Radamacher, Walsh, and Haar Systems are defined and studied. The Fourier Transform is introduced.
The study of Sequence Spaces is the basic and natural setting for the study of Functional Analysis. Banach Sequence Spaces with continuous coordinates (K-spaces) are fundamental here. The concepts of Sectional Convergence (AK) and Cesaro Sectional Convergence (SigmaK), as well as more general sectional density (AD), are examined in relation to bv-invariance and q-invariance.
Linear Operators and Linear Functionals are studied. Then Operator Spaces, Topological Duals, and Second Duals of Normed Spaces are considered. Lebesgue L(p) Spaces are defined and studied. The theorems of Hahn–Banach Extension, Baire Category, Riesz Representation, Open Mapping, Closed Graph, and Banach Fixed Points are all proven.
Frechet K-spaces (FK-spaces) are introduced. They are more general than BK-spaces and are more suited to the study of Topological Sequence Spaces. All Matrix Mappings between FK-spaces are continuous. Alpha, Beta, and Sigma duality between Sequence Spaces are considered and the more general Multiplier Spaces are studied. An application to Matrix Mechanics of Quantum Theory is discussed.
Metric Spaces, Normed Spaces, and Banach Spaces are investigated. Topological concepts of Open and Closed sets, Convergence, Continuity, Compactness, Completeness, and Total Boundedness are studied. The Stone–Weierstrass Approximation Theorem is proven.
This chapter starts with definitions and basic properties of Measurable Functions. The Lebesgue Integral is systematically developed; first for Simple Functions, then for Nonnegative Functions, and finally for the General Case. Egorov's Theorem is proven. The important Lebesgue Limit Theorems that distinguish Lebesgue Integration from Riemann and Stieltjes Integration are discussed and proven.
Summability Theory deals with the assignment a sum to a divergent series or a limit to a divergent sequence. The basics of Summability Theory are given in this chapter. Cesaro Summability is studied in some detail. It is an example of a Matrix Summability Method. The Silverman–Toeplitz Theorem, which characterizes Matrix Summability Methods that sum all convergent series, is proven. Also, the Abel Summability Method, which is not a Matrix Summability Method, is studied.
Outer, Inner, and Lebesgue Measure are defined and systematically studied; first for (n-dimensional) intervals, then for finite and countable union of intervals, then for open and closed sets, and finally for general Lebesgue Measurable sets in Euclidean Spaces. The Approximation Theorem and the Caratheodory Characterization of Measurability are proven. Borel sets are studied and examples are given of Nonmeasurable Sets, as well as Measurable Sets which are not Borel.
Functional analysis deals with infinite-dimensional spaces. Its results are among the greatest achievements of modern mathematics and it has wide-reaching applications to probability theory, statistics, economics, classical and quantum physics, chemistry, engineering, and pure mathematics. This book deals with measure theory and discrete aspects of functional analysis, including Fourier series, sequence spaces, matrix maps, and summability. Based on the author's extensive teaching experience, the text is accessible to advanced undergraduate and first-year graduate students. It can be used as a basis for a one-term course or for a one-year sequence, and is suitable for self-study for readers with an undergraduate-level understanding of real analysis and linear algebra. More than 750 exercises are included to help the reader test their understanding. Key background material is summarized in the Preliminaries.