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Functional Analysis
Present day research in partial differential equations uses a lot of functional analytic techniques. This book treats these methods concisely, in one volume, at the graduate level. It introduces distribution theory (which is fundamental to the study of partial differential equations) and Sobolev spaces (the natural setting in which to find generalized solutions of PDE). Examples, counter-examples, and exercises are included.
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December 5, 2025
Despite the name, this is one of the better written PDE books out there. It treats partial differential equations as applications of functional analysis, and its handling of function spaces and operators is correspondingly rigorous. While this book came about earlier, it should be used in conjunction with Kesavans' Measure and Integration and Functional Analysis. When reading those two books, readers will find that there are certain results not normally available in books on Measure Theory of Functional Analysis, but which are used in this book. The only thing missing is that the new editions should directly refer to the said results in the other two books, rather than one frantically searching for the actual statement based on what one needs. The authour also has a NPTEL course, avaible on YouTube, to go with the book, but the quality of those I can't judge for I haven't watched the.
This (3rd) edition is the best one for me, the earlier editions, especially the first one Topics in Functional Analysis and Applications, have many many typos and omit several important topics. The latest (4th) edition retains all the typos and mistakes of the third, the main change is the typesetting, which is more "modern" and non-compact. Also, the price. Up 33% from 300 to 400. NO typesetting is worth that.
Still, there are errors in the books. Notably, in the section related to the Trace Theorem. One should refer to Krantz's (he does write on almost everything I need) Partial Differential Equations and Complex Analysis and McLean's Strongly Elliptic Systems and Boundary Integral Equations. The section on Sobolov Spaces should be supplemented with Real Analysis: Modern Techniques and Their Applications and the boundary Sobolev Spaces can be better learned from J-Holomorphic Curves and Symplectic Topology (COLLOQUIUM PUBLICATIONS (despite the Scary sounding title, I still haven't dared to try comprehend what title is about, the relevant appendix is self contanined and rigorous in treatment). Keeping a standard PDE reference such as Evans’ Partial Differential Equations at hand is also advisable, as one can easily get lost in the functional-analytic machinery and techniccalities.
In summaryt this is a good textbook for a secon course on PDEs with a Generlized Functions viewpoint. Moreso for people primarily working on hard analysis who view PDEs as a tool for their work and those who are fed up with the general lack of rigour in PDE books as a whole.
This (3rd) edition is the best one for me, the earlier editions, especially the first one Topics in Functional Analysis and Applications, have many many typos and omit several important topics. The latest (4th) edition retains all the typos and mistakes of the third, the main change is the typesetting, which is more "modern" and non-compact. Also, the price. Up 33% from 300 to 400. NO typesetting is worth that.
Still, there are errors in the books. Notably, in the section related to the Trace Theorem. One should refer to Krantz's (he does write on almost everything I need) Partial Differential Equations and Complex Analysis and McLean's Strongly Elliptic Systems and Boundary Integral Equations. The section on Sobolov Spaces should be supplemented with Real Analysis: Modern Techniques and Their Applications and the boundary Sobolev Spaces can be better learned from J-Holomorphic Curves and Symplectic Topology (COLLOQUIUM PUBLICATIONS (despite the Scary sounding title, I still haven't dared to try comprehend what title is about, the relevant appendix is self contanined and rigorous in treatment). Keeping a standard PDE reference such as Evans’ Partial Differential Equations at hand is also advisable, as one can easily get lost in the functional-analytic machinery and techniccalities.
In summaryt this is a good textbook for a secon course on PDEs with a Generlized Functions viewpoint. Moreso for people primarily working on hard analysis who view PDEs as a tool for their work and those who are fed up with the general lack of rigour in PDE books as a whole.
Displaying 1 of 1 review