# e-book Applied Functional Analysis, Second Edition

A novel, practical introduction to functional analysis. In the twenty years since the first edition of Applied Functional Analysis was published.

Jean-Pierre Aubin updates his popular reference on functional analysis with new insights and recent discoveries-adding three new chapters on set-valued analysis and convex analysis, viability kernels and capture basins, and first-order partial differential equations. He presents, for the first time at an introductory level, the extension of differential calculus in the framework of both the theory of distributions and set-valued analysis, and discusses their application for studying boundary-value problems for elliptic and parabolic partial differential equations and for systems of first-order partial differential equations.

To keep the presentation concise and accessible, Jean-Pierre Aubin introduces functional analysis through the simple Hilbertian structure. He seamlessly blends pure mathematics with applied areas that illustrate the theory, incorporating a broad range of examples from numerical analysis, systems theory, calculus of variations, control and optimization theory, convex and nonsmooth analysis, and more. Finally, a summary of the essential theorems as well as exercises reinforcing key concepts are provided.

Applied Functional Analysis, Second Edition is an excellent and timely resource for both pure and applied mathematicians. General Banach spaces are more complicated than Hilbert spaces, and cannot be classified in such a simple manner as those. In particular, many Banach spaces lack a notion analogous to an orthonormal basis.

That is, we require.

In Banach spaces, a large part of the study involves the dual space : the space of all continuous linear maps from the space into its underlying field, so-called functionals. A Banach space can be canonically identified with a subspace of its bidual, which is the dual of its dual space. The corresponding map is an isometry but in general not onto. A general Banach space and its bidual need not even be isometrically isomorphic in any way, contrary to the finite-dimensional situation.

### ISBN 10: 0387905278

This is explained in the dual space article. Also, the notion of derivative can be extended to arbitrary functions between Banach spaces.

The uniform boundedness principle or Banach—Steinhaus theorem is one of the fundamental results in functional analysis. Together with the Hahn—Banach theorem and the open mapping theorem , it is considered one of the cornerstones of the field. In its basic form, it asserts that for a family of continuous linear operators and thus bounded operators whose domain is a Banach space , pointwise boundedness is equivalent to uniform boundedness in operator norm.

The theorem was first published in by Stefan Banach and Hugo Steinhaus but it was also proven independently by Hans Hahn.

Theorem Uniform Boundedness Principle. Let X be a Banach space and Y be a normed vector space. Suppose that F is a collection of continuous linear operators from X to Y.

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If for all x in X one has. There are many theorems known as the spectral theorem , but one in particular has many applications in functional analysis. Theorem:  Let A be a bounded self-adjoint operator on a Hilbert space H. This is the beginning of the vast research area of functional analysis called operator theory ; see also the spectral measure. There is also an analogous spectral theorem for bounded normal operators on Hilbert spaces.

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The Hahn—Banach theorem is a central tool in functional analysis. It allows the extension of bounded linear functionals defined on a subspace of some vector space to the whole space, and it also shows that there are "enough" continuous linear functionals defined on every normed vector space to make the study of the dual space "interesting". The open mapping theorem , also known as the Banach—Schauder theorem named after Stefan Banach and Juliusz Schauder , is a fundamental result which states that if a continuous linear operator between Banach spaces is surjective then it is an open map.

More precisely,: .

The proof uses the Baire category theorem , and completeness of both X and Y is essential to the theorem. The closed graph theorem states the following: If X is a topological space and Y is a compact Hausdorff space , then the graph of a linear map T from X to Y is closed if and only if T is continuous. Most spaces considered in functional analysis have infinite dimension.

To show the existence of a vector space basis for such spaces may require Zorn's lemma. However, a somewhat different concept, Schauder basis , is usually more relevant in functional analysis. Many very important theorems require the Hahn—Banach theorem , usually proved using axiom of choice , although the strictly weaker Boolean prime ideal theorem suffices. The Baire category theorem , needed to prove many important theorems, also requires a form of axiom of choice.

## Functional analysis - Wikipedia

Functional analysis in its present form [update] includes the following tendencies:. From Wikipedia, the free encyclopedia. For the assessment and treatment of human behavior, see Functional analysis psychology. Main article: Banach-Steinhaus theorem. Main article: Spectral theorem. Main article: Hahn—Banach theorem. Main article: Open mapping theorem functional analysis. Main article: Closed graph theorem.