Dvoretzky–Rogers theorem

Remarks

Taking $c_n = 1/n$ gives a series in $X$ that converges unconditionally but not absolutely. 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.

Going deeper

For a student

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.