Book contents
- Frontmatter
- Contents
- Preface
- Acknowledgements
- Picture: closing at the main lecture room
- 1 GEOMETRICAL METHODS
- 2 HOMOLOGICAL METHODS
- Yet another proof of Sobczyk's theorem
- The category of exact sequences of Banach spaces
- Extension problems for C(K) spaces and twisted sums
- Palamodov's questions from Homological methods in the theory of locally convex spaces
- 3 TOPOLOGICAL METHODS
- 4 OPERATOR THEORY METHODS
- 5 FUNCTION SPACE METHODS
- List of participants
- Picture: some like it fun
Palamodov's questions from Homological methods in the theory of locally convex spaces
Published online by Cambridge University Press: 04 May 2010
- Frontmatter
- Contents
- Preface
- Acknowledgements
- Picture: closing at the main lecture room
- 1 GEOMETRICAL METHODS
- 2 HOMOLOGICAL METHODS
- Yet another proof of Sobczyk's theorem
- The category of exact sequences of Banach spaces
- Extension problems for C(K) spaces and twisted sums
- Palamodov's questions from Homological methods in the theory of locally convex spaces
- 3 TOPOLOGICAL METHODS
- 4 OPERATOR THEORY METHODS
- 5 FUNCTION SPACE METHODS
- List of participants
- Picture: some like it fun
Summary
Abstract. In his seminal work from 1971, V. P. Palamodov introduced methods from category theory and homological algebra to the theory of locally convex spaces. These methods shed new light on many classical topics and led to many applications in analysis, e.g. for partial differential operators. The final section of Palamodov's article posed eight open problems. We will try to explain the motivation for these questions as well as their solutions which show that Palamodov's problems had been a bit too optimistic.
THE EMPEROR'S NEW CLOTHES?
Usually, homological methods do not solve a problem at once, but they may tell rather precisely what has to be done for the solution. Once knowing the solution it is often possible to give a presentation which avoids the abstract homological methods and which may then look even ingenious. Perhaps, this is one of the reasons why seemingly many mathematicians dislike homological tools: it looks as if they were superfluous. Another reason certainly is that these general tools, which found applications in all parts of mathematics from algebra and topology to algebraic geometry as well as – and this is the concern of the present article – to functional analysis, have to be formulated in a very general abstract language. One might get the feeling that only trivialities can be true in such a generality, and like in Andersen's “The emperor's new clothes” one waits for a child telling that he is naked. Indeed, in his book on algebra [11, page 105], Serge Lang posed the exercise: “Take any book on homological algebra, and prove all the theorems without looking at the proofs given in that book”.
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- Methods in Banach Space Theory , pp. 169 - 182Publisher: Cambridge University PressPrint publication year: 2006
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