Book contents
- Frontmatter
- Contents
- Introduction
- Notation
- 1 Unconditional and Absolute Summability in Banach Spaces
- 2 Fundamentals of p-Summing Operators
- 3 Summing Operators on Lp-Spaces
- 4 Operators on Hilbert Spaces and Summing Operators
- 5 p-Integral Operators
- 6 Trace Duality
- 7 2–Factorable Operators
- 8 Ultraproducts and Local Reflexivity
- 9 p-Factorable Operators
- 10 (q, p)-Summing Operators
- 11 Type and Cotype: The Basics
- 12 Randomized Series and Almost Summing Operators
- 13 K-Convexity and B-Convexity
- 14 Spaces with Finite Cotype
- 15 Weakly Compact Operators on C(K)-Spaces
- 16 Type and Cotype in Banach Lattices
- 17 Local Unconditionality
- 18 Summing Algebras
- 19 Dvoretzky's Theorem and Factorization of Operators
- References
- Author Index
- Subject Index
1 - Unconditional and Absolute Summability in Banach Spaces
Published online by Cambridge University Press: 21 October 2009
- Frontmatter
- Contents
- Introduction
- Notation
- 1 Unconditional and Absolute Summability in Banach Spaces
- 2 Fundamentals of p-Summing Operators
- 3 Summing Operators on Lp-Spaces
- 4 Operators on Hilbert Spaces and Summing Operators
- 5 p-Integral Operators
- 6 Trace Duality
- 7 2–Factorable Operators
- 8 Ultraproducts and Local Reflexivity
- 9 p-Factorable Operators
- 10 (q, p)-Summing Operators
- 11 Type and Cotype: The Basics
- 12 Randomized Series and Almost Summing Operators
- 13 K-Convexity and B-Convexity
- 14 Spaces with Finite Cotype
- 15 Weakly Compact Operators on C(K)-Spaces
- 16 Type and Cotype in Banach Lattices
- 17 Local Unconditionality
- 18 Summing Algebras
- 19 Dvoretzky's Theorem and Factorization of Operators
- References
- Author Index
- Subject Index
Summary
The Dvoretzky - Rogers Theorem
Recall that a sequence (xn) in a normed space is absolutely summable if Σn ∥xn∥ < ∞, and is unconditionally summable if Σn xσ(n) converges, regardless of the permutation σ of the indices. It is traditional to say that the series Σn xn is absolutely (unconditionally) convergent if the sequence (xn) is absolutely (unconditionally) summable.
A theorem of Dirichlet from elementary analysis asserts that a scalar sequence is absolutely summable precisely when it is unconditionally summable. Simple natural adjustments to the proof show that this theorem extends to the setting of any finite dimensional normed space.
What happens in infinite dimensional spaces? Without completeness we can get nowhere.
1.1 Proposition: A normed space is a Banach space if and only if every absolutely summable sequence is unconditionally summable.
This elementary old standby finds frequent use in proofs of completeness, and a brief indication of its proof is worthy of our attention.
Proof. To show completeness we need to prove that every Cauchy sequence (xn) is convergent. For this it suffices to find a convergent subsequence, a task which is not difficult since any ‘sufficiently rapid’ Cauchy subsequence will do the trick. For example, choose an increasing sequence of positive integers (nk) so that if yk = xnk+1 − xnk, then ∥yk∥ ≤ 2−k. As (yk) is absolutely summable, it is (unconditionally) summable. The convergence of (xnk) now follows from the identity xn1 + y1 + … + yk = xnk+1.
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- Chapter
- Information
- Absolutely Summing Operators , pp. 1 - 30Publisher: Cambridge University PressPrint publication year: 1995