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2006-10-27 Seminar

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> //**This lecture will be held on Friday on 15.00 in room B3-38**//


This lecture will be held on Friday on 15.00 in room B3-38

prof. dr hab. Grzegorz Lewicki (Institute of Mathematics of Jagiellonian University)
Spaces with maximal projection constant

Let $$ X $$ be a real Banach space and $$ Y\subset X $$ its closed subspace. Denote by $$ \mathcal{P}(X,Y) $$ the set of all linear and continuous projections from $$ X $$ on $$ Y $$. Let $$ \lambda(Y,X)=\inf\{\|P\|:P\in\mathcal{P}(X,Y)\} $$. For $$ n $$-dimensional Banach space $$ Y $$ let $$ \lambda(Y) = \sup \{ \lambda(Y,X): Y \subset X \} $$. The constant $$ \lambda(Y) $$ is called the absolute projection constant. It is a well-known fact that for $$ n $$-dimensional Banach space $$ Y $$, $$ \lambda(Y) = \lambda(Y, l_{\infty}) $$. For fixed $$ n, N \in \mathbb{N} $$, $$ n < N $$ let $$ \lambda^N_n(Y) = \sup\{ \lambda(Y) : Y \subset l_{\infty}^{(N)},\, \dim(Y) = n\} $$. During the lecture I will present some results and open problems concerning effective determining $$ n $$-dimensional spaces $$ Y\subset l_{\infty}^{(N)} $$ such that $$ \lambda(Y, l_{\infty}^{(N)}) = \lambda^N_n $$.