Banach spaces in which a~theorem of Orlicz is not true

S. A. Rakov

Matematičeskie zametki, Tome 14 (1973) no. 1, pp. 101-106.

Résumé

Let the Banach space $X$ be such that for every numerical sequencet $l_n\searrow0$ there exists in $X$ an unconditionally convergent series $\Sigma x_n$, the terms of which are subject to the condition $\|x_n\|=t_n$ ($n=1,2,\dots$). Then $$\sup_n\inf_{X_n}d(X_n,l_\infty^{(n)})\infty,$$ where $X_n$ ranges over all the $n$-dimensional subspaces of $X$.