Topological Vector Spaces and Their Applications

1. Front Matter

   Pages i-x

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2. Introduction to the theory of topological vector spaces

   • V. I. Bogachev, O. G. Smolyanov

   Pages 1-100

3. Methods of constructing topological vector spaces

   • V. I. Bogachev, O. G. Smolyanov

   Pages 101-152

4. Duality

   • V. I. Bogachev, O. G. Smolyanov

   Pages 153-242

5. Differential calculus

   • V. I. Bogachev, O. G. Smolyanov

   Pages 243-310

6. Measures on linear spaces

   • V. I. Bogachev, O. G. Smolyanov

   Pages 311-418

7. Back Matter

   Pages 419-456

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This book gives a compact exposition of the fundamentals of the theory of locally convex topological vector spaces. Furthermore it contains a survey of the most important results of a more subtle nature, which cannot be regarded as basic, but knowledge which is useful for understanding applications. Finally, the book explores some of such applications connected with differential calculus and measure theory in infinite-dimensional spaces. These applications are a central aspect of the book, which is why it is different from the wide range of existing texts on topological vector spaces. Overall, this book develops differential and integral calculus on infinite-dimensional locally convex spaces by using methods and techniques of the theory of locally convex spaces.                                                                                                    

The target readership includes mathematicians and physicists whose research is related to infinite-dimensional analysis.

• 46A03, 58C20, 28C20
• Topological vector space
• Locally convex space
• Differentiation in infinite-dimensional spaces
• Integration in infinite-dimensional spaces
• Measures on infinite-dimensional spaces

“The book under review presents an excellent modern treatment of topological linear spaces. Moreover, in contrast to existing monographs on this topic it adds material on applications that are not covered elsewhere. … The book is well written and elucidates basic concepts with a large list of examples.” (Jan Hamhalter, Mathematical Reviews, November, 2017)



“This is indeed a good book, well written, that includes much useful material. The basic theory is presented in a clear, understandable way. Moreover, many recent, important, more specialized results are also included with precise references. This book is recommendable for analysts interested in the modern theory of locally convex spaces and its applications, and especially for those mathematicians who might use differentiation theory on infinite-dimensional spaces or measure theory on topological vector spaces.” (José Bonet, zbMATH 1378.46001, 2018)

• Department of Mechanics and Mathematics, Moscow State University, Moscow, Russia

  V.I. Bogachev, O.G. Smolyanov

Vladimir Bogachev, born in 1961, Professor at the Department of Mechanics and Mathematics of Lomonosov Moscow State University and at the Faculty of Mathematics of the Higher School of Economics (Moscow, Russia) is an expert in measure theory and infinite-dimensional analysis and the author of more than 200 papers and 12 monographs, including his famous two-volume treatise "Measure theory" (Springer, 2007), "Gaussian measures" (AMS, 1997), "Differentiable measures and the Malliavin calculus" (AMS, 2010), "Fokker-Planck-Kolmogorov equations" (AMS, 2015), and others. An author with a high citation index (h=31 with more than 4700 citations according to the Google Scholar), Vladimir Bogachev solved several long-standing problems in measure theory and Fokker-Planck-Kolmogorov equations. 

Oleg Smolyanov, born in 1938, Professor at the Department of Mechanics and Mathematics of Lomonosov Moscow State University is an expert in topological vector spacesand infinite-dimensional analysis and author of more than 200 papers and 5 monographs. Oleg Smolyanov solved several long-standing problems in the theory of topological vector spaces.

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