# s-numbers of projections in Banach spaces

Mascioni, V (1989). s-numbers of projections in Banach spaces. Israel Journal of Mathematics, 67(1):82-94.

## Abstract

Given ans-number sequences te {h, x, y, c, d, a, Γ}, we find a characterization of the following property of a Banach spaceX:(P s). There is a constantC>0 such that, for anyn-dimensional subspaceE ofX, we can find a projectionP fromX ontoE with sup k ks k(P)≦Cn. As an application, we prove thatX has weak type 2 if and only ifX is finite dimensionally norming, thus answering a question of Casazza and Shura. Weak Hilbert spaces are also characterized in a new way, the main tool in the proof being a characterization of weak cotype 2 by means of projections. The latter is applied to the study of U.A.P., too.

## Abstract

Given ans-number sequences te {h, x, y, c, d, a, Γ}, we find a characterization of the following property of a Banach spaceX:(P s). There is a constantC>0 such that, for anyn-dimensional subspaceE ofX, we can find a projectionP fromX ontoE with sup k ks k(P)≦Cn. As an application, we prove thatX has weak type 2 if and only ifX is finite dimensionally norming, thus answering a question of Casazza and Shura. Weak Hilbert spaces are also characterized in a new way, the main tool in the proof being a characterization of weak cotype 2 by means of projections. The latter is applied to the study of U.A.P., too.

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