Speaker
Description
All spaces below are not Hilbert spaces. Given two finite dimensional subspaces $L, K$ of a normed space $X$ we call $K$ orthogonal to $L$ if for every unit vector in $K$ the distance of this vector to $L$ is $1$.
This usually does not mean that $L$ is orthogonal to $K$.
We consider the following questions: 1) Let $E, F$ are two finite dimensional subspaces of a normed space $X$ and let $dim F= dim E+m$. Can we always find a subspace $K$ in $F$ such that $E$ is orthogonal to $K$? 2) Can we always find a subspace $K$ in $F$ such that $K$ is orthogonal to $E$? 3) Can we always choose $K$ of dimension $m$? 4) If not, what is the maximal possible dimension?
These questions seem to be considered 60-80 years ago, and in fact, some of them were answered (by Krein--Krasnoselski--Milman). But it looks like that some of these questions were overlooked...
