Read e-book online Applied Abstract Analysis PDF

By Jean-Pierre Aubin, Bernard Cornet, Herve Moulin

ISBN-10: 0471021466

ISBN-13: 9780471021469

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Proof. ) According to Proposition 2, every normed space is a metric space. The study of normed spaces constitutes a part of what is called functional analysis. The structure of a normed space, combining the structures of a vector space and of a metric space, has a large role both in mathematical theory and in applications. Complete normed spaces, in particular, play a very important role. Definition 2. A “ Banach space” is a complete normed vector space. We see later how to characterize the norms on a vector space.

They possess many more properties than Banach spaces or complete metric spaces. In analysis, we try whenever possible to work within the framework of Hilbert spaces in order to obtain more results and with less difficulty. This is what will be done in [AFA]. ■ Remark 3. In fact, the fundamental property of Hilbert spaces is the linearity of the Frechet derivative of the associated quadratic function (p, defined by (23)

Hilbert spaces have a richer structure than Banach spaces or, a fortiori, than metric spaces. They possess many more properties than Banach spaces or complete metric spaces. In analysis, we try whenever possible to work within the framework of Hilbert spaces in order to obtain more results and with less difficulty. This is what will be done in [AFA]. ■ Remark 3. In fact, the fundamental property of Hilbert spaces is the linearity of the Frechet derivative of the associated quadratic function (p, defined by (23)

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Applied Abstract Analysis by Jean-Pierre Aubin, Bernard Cornet, Herve Moulin


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