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
The category of exact sequences of Banach 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
To Atenea.
Consider nature's magnificent foresight
spreading seeds of madness everywhere.
If mortals would refrain from no matter which contact
with wisdom, even the old age would not exist.
Life is not different from a dreaming game
whose greatest gifts come to us through craziness.
Consider nature's magnificent foresight
in making the heart be always right.
The purpose of this paper is to lay the foundations for the construction of the category of exact sequences of Banach spaces; the construction for quasi-Banach spaces is analogous and thus we omit it. The construction of a category associated to a theory, in addition to its intrinsic value, provides the right context to study, among others, isomorphic and universal objects. In our particular case, let us describe a couple of phenomena often encountered when working with exact sequences of Banach spaces for which the categorical approach provides rigorous explanations.
If one “multiplies” an exact sequence 0 → Y → X → Z → 0 by the left (resp. right) by a given space E, the resulting exact sequence 0 → E ⊕ Y → E ⊕ X → Z → 0 (resp. 0 → Y → X ⊕ E → Z ⊕ E → 0) is “the same”. And this holds despite the fact that the original and the “multiplied” sequences are not equivalent under any known definition. The categorical approach provides the simplest explanation: the two sequences are isomorphic objects in the category.
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- Chapter
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- Methods in Banach Space Theory , pp. 139 - 158Publisher: Cambridge University PressPrint publication year: 2006
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