Linear Functional Analysis

Authors: Alt, Hans Wilhelm
A complete, self-contained introduction to linear functional analysis
Includes topics such as operator theory, distributions, Sobolev spaces
Features many solved exercises

Even the simplest mathematical abstraction of the phenomena of reality­ the real line-can be regarded from different points of view by different mathematical disciplines. For example, the algebraic approach to the study of the real line involves describing its properties as a set to whose elements we can apply" operations," and obtaining an algebraic model of it on the basis of these properties, without regard for the topological properties. On the other hand, we can focus on the topology of the real line and construct a formal model of it by singling out its" continuity" as a basis for the model.

Functional analysis has become a sufficiently large area of mathematics that it is possible to find two research mathematicians, both of whom call themselves functional analysts, who have great difficulty understanding the work of the other. The common thread is the existence of a linear space with a topology or two (or more).

This book provides the foundations for a rigorous theory of functional analysis with bicomplex scalars. It begins with a detailed study of bicomplex and hyperbolic numbers, and then defines the notion of bicomplex modules.

This introduction to the ideas and methods of linear functional analysis shows how familiar and useful concepts from finite-dimensional linear algebra can be extended or generalized to infinite-dimensional spaces.

Originally, functional analysis was that branch of mathematics capable of investigating in an abstract way a series of linear mathematical models from science. The study of these linear models—in fact, only first approximations of real models—proved insufficient, so the theory had to be extended to be able to describe the nonlinear phenomena themselves. In this way nonlinear functional analysis was born and continues to develop, becoming a vast and fascinating field of mathematics, with deep applications to increasingly complex problems in physics, biology, chemistry, and economics.