The Hahn-Banach Theorems. Introduction to the Theory of Conjugate Convex Functions. Unbounded Operators. Characterization of Surjective Operators. Weak Topologies. Reflexive Spaces.
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Although there are many books on functional analysis and many on PDEs, this is the first to cover both of these closely connected topics. Moreover, the wealth of exercises and additional material presented, leads the reader to the frontier of research. The English version is a welcome addition to this list. The first part of the text deals with abstract results in FA and operator theory. The second part is concerned with the study of spaces of functions of one or more real variables having specific differentiability properties, e.
The Sobolev spaces occur in a wide range of questions, both in pure and applied mathematics, appearing in linear and nonlinear PDEs which arise, for example, in differential geometry, harmonic analysis, engineering, mechanics, physics etc.
In fact I would recommend this over any other source to any beginning graduate student. Its a bible for the field of research. It is well-written and I can wholeheartedly recommend it to both students and teachers.
I wholeheartedly recommend this book both as a textbook, as well as for independent study. It has seen translations into numerous languages and the Springer edition was especially anticipated, as it announced a number of practice exercises following each chapter.
I can honestly say that it was well worth the wait.
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Functional Analysis, Sobolev Spaces and Partial Differential Equations