Optimization and Control Theory Department: 2006-10-27 Seminar

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 $\lambda(Y, l_{\infty}^{(N)}) = \lambda^N_n$ .