Functional analysis studies the algebraic, geometric, and topological structures of spaces and operators that underlie many classical problems. Individual functions satisfying specific equations are replaced by classes of functions and transforms that are determined by the particular problems at hand. This book presents the basic facts of linear functional analysis as related to fundamental aspects of mathematical analysis and their applications. The exposition avoids unnecessary terminology and generality and focuses on showing how the knowledge of these structures clarifies what is essential in analytic problems.

The material in the first part of the book can be used for an introductory course on functional analysis, with an emphasis on the role of duality. The second part introduces distributions and Sobolev spaces and their applications. Convolution and the Fourier transform are shown to be useful tools for the study of partial differential equations. Fundamental solutions and Green's functions are considered and the theory is illustrated with several applications. In the last chapters, the Gelfand transform for Banach algebras is used to present the spectral theory of bounded and unbounded operators, which is then used in an introduction to the basic axioms of quantum mechanics.

The presentation is intended to be accessible to readers whose backgrounds include basic linear algebra, integration theory, and general topology. Almost exercises will help the reader in better understanding the concepts employed. Overall, this is an interesting and well-written book, covering a lot of material. I found it very readable The discussion is well motivated mathematically there are few applications to other areas and full details of proofs are given, so it is fairly self-contained.

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## School of Mathematics and Statistics (SoMaS)

Rynne, Mathematical Reviews. Advanced search. Abstract: Functional analysis studies the algebraic, geometric, and topological structures of spaces and operators that underlie many classical problems. Volume: Publication Month and Year: Copyright Year: Page Count: Cover Type: Hardcover.

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## Linear Functional Analysis by Martin A. Youngson

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Linear Functional Analysis by Martin A. This book provides an introduction to the ideas and methods of linear fu- tional analysis at a level appropriate to the? The prerequisites for reading it are a standard undergraduate knowledge of linear algebra and real analysis including the t- ory of metric spaces. Part of the development of functional analys This book provides an introduction to the ideas and methods of linear fu- tional analysis at a level appropriate to the?

Part of the development of functional analysis can be traced to attempts to? Often, the appropriate setting turned out to be a vector space of real or complex-valued functions de? In general, such a v- tor space is in?

This leads to di? For example, in general in?

### Second Edition

Nevertheless, on the spaces of most interest to us there is often a norm which extends the idea of the length of a vector to a somewhat more abstract setting. Since a norm on a vector space gives rise to a metric on the space, it is now possible to do analysis in the space.

As real or complex-valued functions are often called functionals, the term functional analysis came to be used for this topic. We now brie? Get A Copy. Paperback , pages. More Details Original Title. Friend Reviews. To see what your friends thought of this book, please sign up.

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