A constructive proof of the Dvoretzky--Rogers Theorem in $\ell_{p}$ spaces

Daniel Pellegrino, Janiely Silva

Published 2020-06-13Version 1

The Dvoretzky--Rogers Theorem asserts that in every infinite-dimensional Banach space $X$ there exists an unconditionally convergent series $ {\textstyle\sum}x^{(j)}$ such that ${\textstyle\sum}\Vert x^{(j)}\Vert^{^{2-\varepsilon}}=\infty$ for all $\varepsilon>0.$ Their proof is non-constructive and, according to the literature, the case $X=\ell_{1}$ is critical. A constructive proof when $X=$ $\ell_{1}$ in the particular case $\varepsilon\leq1$ was obtained by MacPhail in 1947. However, to the best of the authors' knowledge there is no explicit construction of a series satisfying the whole statement of the Dvoretzky--Rogers theorem for $X=\ell_{1}$ (and even for $\ell_{p}$ with $p\leq2$; the case $p>2$ is trivial). In this note we provide an example of such series for $1\leq p \leq 2$. Our approach rests in a suitable handling of certain special matrices that date back to the works of Toeplitz.