1 - Unconditional and Absolute Summability in Banach Spaces

Summary

The Dvoretzky - Rogers Theorem

Recall that a sequence (x_n) in a normed space is absolutely summable if Σ_n ∥x_n∥ < ∞, and is unconditionally summable if Σ_n x_σ(n) converges, regardless of the permutation σ of the indices. It is traditional to say that the series Σ_n x_n is absolutely (unconditionally) convergent if the sequence (x_n) 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 (x_n) 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 (n_k) so that if y_k = x_nk+1 − x_nk, then ∥y_k∥ ≤ 2^−k. As (y_k) is absolutely summable, it is (unconditionally) summable. The convergence of (x_nk) now follows from the identity x_n1 + y₁ + … + y_k = x_nk+1.