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A theory is the more impressive, the simpler are its premises, the more distinct are the things it connects, and the broader is its range of applicability. Albert Einstein There are two different ways of teaching mathematics, namely, (i) the systematic way, and (ii) the application-oriented way. More precisely, by (i), I mean a systematic presentation of the material governed by the desire for mathematical perfection and completeness of the results. In contrast to (i), approach (ii) starts out from the question "What are the most important applications?" and then tries to answer this question as quickly as possible. Here, one walks directly on the main road and does not wander into all the nice and interesting side roads. The present book is based on the second approach. It is addressed to undergraduate and beginning graduate students of mathematics, physics, and engineering who want to learn how functional analysis elegantly solves mathematical problems that are related to our real world and that have played an important role in the history of mathematics. The reader should sense that the theory is being developed, not simply for its own sake, but for the effective solution of concrete problems. viii Preface This introduction to functional analysis is divided into the following two parts: Part I: Applications to mathematical physics (the present AMS Vol. 108); Part II: Main principles and their applications (AMS Vol. 109).

Inhaltsverzeichnis

Frontmatter

1. Banach Spaces and Fixed-Point Theorems

Abstract
In a Banach space, the so-called norm
$$ \parallel u\parallel = nonnegativenumber \hfill \\ $$
is assigned to each element u. This generalizes the absolute value |u of a real number u. The norm can be used in order to define the convergence
$$ \mathop {\lim }\limits_{n \to \infty } {u_n} = u \hfill \\ $$
by means of
$$ \mathop {\lim }\limits_{n \to \infty } \parallel {u_n} - u\parallel = 0. \hfill \\ \parallel u\parallel = nonnegativenumber \hfill \\ $$
Eberhard Zeidler

2. Hilbert Spaces, Orthogonality, and the Dirichlet Principle

Abstract
In a famous paper from 1857, Riemann used the Dirichlet principle for the foundation of the theory of complex analytic functions. In 1870 Weierstrass showed that there are variational problems that do not have any solution.1 This way the justification of the Dirichlet principle became an important open problem, which Hilbert solved in 1900.
Eberhard Zeidler

3. Hilbert Spaces and Generalized Fourier Series

Abstract
Let f: ℝ → ℝ be a function of period 2 π.
Eberhard Zeidler

4. Eigenvalue Problems for Linear Compact Symmetric Operators

Abstract
In this chapter we want to study the following eigenvalue problem: \(Au\, = \lambda u,\;\quad u \in X,\quad \lambda \in \mathbb{K},\quad u \ne 0,\)on the Hilbert space X over K,along with applications to integral equations and boundary-value problems.
Eberhard Zeidler

5. Self-Adjoint Operators, the Friedrichs Extension, and the Partial Differential Equations of Mathematical Physics

Abstract
In this Chapter we want to study the following problems in a Hilbert space X, wehere A: D(A) ⊆ XX is a linear symmetric operator that has additional properties to be discussed ahead (cf. also Figure 5.1).
Eberhard Zeidler

Backmatter

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