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2019 | Buch

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Functional Analysis for the Applied Sciences

verfasst von: Gheorghe Moroşanu

Verlag: Springer International Publishing

Buchreihe : Universitext

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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Über dieses Buch

This advanced graduate textbook presents main results and techniques in Functional Analysis and uses them to explore other areas of mathematics and applications. Special attention is paid to creating appropriate frameworks towards solving significant problems involving diﬀerential and integral equations. Exercises at the end of each chapter help the reader to understand the richness of ideas and methods offered by Functional Analysis. Some of the exercises supplement theoretical material, while others relate to the real world. This textbook, with its friendly exposition, focuses on different problems in physics and other applied sciences and uniquely provides solutions to most of the exercises. The text is aimed toward graduate students and researchers in applied mathematics, physics, and neighboring fields of science.

Inhaltsverzeichnis

Frontmatter

Chapter 1. Introduction

Abstract

This chapter comprises definitions, notation, and basic results related to set theory, real and complex numbers, and linear spaces.

Gheorghe Moroşanu

Chapter 2. Metric Spaces

Abstract

Metric spaces offer a sufficiently large framework for most of the problems we discuss in this book.

Gheorghe Moroşanu

Chapter 3. The Lebesgue Integral and Lp Spaces

Abstract

In this chapter we discuss Lebesgue measurable sets, Lebesgue measurable functions, Lebesgue integration, and L ^p spaces. These spaces, equipped with appropriate norms, are significant examples of Banach spaces.

Gheorghe Moroşanu

Chapter 4. Continuous Linear Operators and Functionals

Abstract

In this chapter we discuss linear operators between linear spaces, but our presentation is restricted at this stage to the space of continuous (bounded) linear operators between normed spaces. When the target space is either $\mathbb {R}$ or $\mathbb {C}$, they are called (continuous linear) functionals and are used to define dual spaces and weak topologies.

Gheorghe Moroşanu

Chapter 5. Distributions, Sobolev Spaces

Abstract

In this chapter we first present test functions, which are then used to introduce scalar distributions. The space ${\mathcal {D}}'(\varOmega )$ of distributions is analyzed in detail and some related applications are discussed: the interpretation of the density of a mass concentrated at a point by means of the Dirac distribution, solving the Poisson equation in ${\mathcal {D}}'(\varOmega )$, solving ordinary differential equations in ${\mathcal {D}}'(\mathbb {R})$, solving the equation of the vibrating string with non-smooth initial displacement function, and the boundary controllability for a problem associated with the same wave equation. We also introduce and discuss Sobolev spaces. In order to introduce vector distributions we shall present in a separate section the Bochner integral for vector functions. Vector distributions and W ^{k, p}(a, b; X) spaces are then presented. These will later be used in solving problems associated with parabolic and hyperbolic PDE’s.

Gheorghe Moroşanu

Chapter 6. Hilbert Spaces

Abstract

Let X be a linear space over $\mathbb {K}$ equipped with a scalar (inner) product (⋅, ⋅) (i.e., X is an inner product space or a generalized Euclidean space, as defined in Chap. 1). As usual, throughout this chapter $\mathbb {K}$ is either $\mathbb {R}$ or $\mathbb {C}$. Define the norm

$$\displaystyle \Vert x\Vert = \sqrt {(x,x)}, \ \ x\in X \, . $$

If (X, ∥⋅∥) is a Banach space (i.e., (X, d) is a complete metric space, where d(x, y) = ∥x − y∥, x, y ∈ X), then X is said to be a Hilbert space. In other words, a Hilbert space is a Banach space (X, ∥⋅∥) whose norm is given by a scalar product.

Gheorghe Moroşanu

Chapter 7. Adjoint, Symmetric, and Self-adjoint Linear Operators

Abstract

Here we first recall the definition of the adjoint of a linear operator and discuss some related results. Then we shall address the case of compact operators A : H → H, where H is a Hilbert space, and present the Fredholm theorem as an application. The last section is devoted to symmetric operators and self-adjoint operators.

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Chapter 8. Eigenvalues and Eigenvectors

Abstract

In this chapter we present the main results regarding eigenvalues and eigenvectors of compact and/or symmetric operators. This includes the Hilbert–Schmidt Theorem and its applications to the main eigenvalue problems for the Laplacian.

Gheorghe Moroşanu

Chapter 9. Semigroups of Linear Operators

Abstract

Let A be an n × n matrix with entries $a_{ij}\in \mathbb {C}$ for all i, j = 1, 2, …, n.

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Chapter 10. Solving Linear Evolution Equations by the Fourier Method

Abstract

In Chap. 9 we used the linear semigroup approach to solve inhomogeneous linear evolution equations. For the same purpose, we use here the Fourier method. More precisely, under appropriate conditions on the linear operators governing such equations, we find the solutions in the form of Fourier series expansions. This approach is based in an essential way on the results discussed in Chap. 8.

Gheorghe Moroşanu

Chapter 11. Integral Equations

Abstract

This chapter is an introduction to the theory of linear Volterra and Fredholm equations. Some aspects related to certain nonlinear extensions are also addressed.

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Chapter 12. Answers to Exercises

Abstract

This chapter provides solutions to almost all exercises proposed at the end of each chapter. The solutions are labeled with the same numbers used for the corresponding exercises. For easy exercises we shall provide hints or just their final solutions. Answers to very easy exercises are left to the reader.

Gheorghe Moroşanu

Backmatter

Titel: Functional Analysis for the Applied Sciences
verfasst von: Gheorghe Moroşanu
Verlag: Springer International Publishing
Electronic ISBN: 978-3-030-27153-4
Print ISBN: 978-3-030-27152-7
DOI: https://doi.org/10.1007/978-3-030-27153-4