---
tag: H924SAD12G
kind: statement
role: theorem
name: Dvoretzky–Rogers theorem
msc: [46B15, 40A05]
uses: [0BDTTM24DP, 2ME9M61TJK, 0W3K3QPEWT]
about: [2ME9M61TJK]
sources:
  - url: https://en.wikipedia.org/wiki/Unconditional_convergence
---

## Statement

Let $X$ be an infinite-dimensional [[Banach space|0BDTTM24DP]].
Let $(c_n)$ be a sequence of nonnegative real numbers with $\sum_n c_n^2 < \infty$.
Then there is a sequence $(x_n)$ in $X$ with $\norm{x_n} = c_n$ for all $n$ such that $\sum_n x_n$ [[converges unconditionally|2ME9M61TJK]].

## Remarks

Taking $c_n = 1/n$ gives a series in $X$ that converges unconditionally but not [[absolutely|0W3K3QPEWT]].
In finite dimensions every unconditionally convergent series converges absolutely.
So a Banach space is finite-dimensional if and only if unconditional and absolute convergence of series coincide in it.
In a Hilbert space, every unconditionally convergent series satisfies $\sum \norm{x_n}^2 < \infty$, so the condition on $(c_n)$ cannot be weakened in general.
The proof uses the Dvoretzky–Rogers lemma, a statement about the geometry of finite-dimensional normed spaces.

## Explanations

### L1 human

In ordinary space, a series that converges in every order also converges when you replace each vector by its length.
The theorem says that this fails in every infinite-dimensional Banach space.
There are always series whose sum does not depend on the order, although the lengths of the terms add up to infinity.

### L2 human

The theorem says that absolute and unconditional convergence agree only in finite dimensions.
It shows that the example of $\sum e_n / n$ in $\ell^2$ is not special to Hilbert space.
Its proof works inside large finite-dimensional subspaces of $X$, a typical method of local Banach space theory.