Geodesic
Banach spaces in which a~theorem of Orlicz is not true
Matematičeskie zametki, Tome 14 (1973) no. 1, pp. 101-106

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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 $X$. 
@article{MZM_1973_14_1_a12,
     author = {S. A. Rakov},
     title = {Banach spaces in which a~theorem of {Orlicz} is not true},
     journal = {Matemati\v{c}eskie zametki},
     pages = {101--106},
     publisher = {mathdoc},
     volume = {14},
     number = {1},
     year = {1973},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_1973_14_1_a12/}
}
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S. A. Rakov. Banach spaces in which a~theorem of Orlicz is not true. Matematičeskie zametki, Tome 14 (1973) no. 1, pp. 101-106. http://geodesic.mathdoc.fr/item/MZM_1973_14_1_a12/