论文标题
关于一组最小预测的维度
On the dimension of the set of minimal projections
论文作者
论文摘要
令$ x $为有限维度的规范空间，让$ y \ subseteq x $为其适当的线性子空间。从$ x $到$ y $的所有最低预测的集合是空间的凸子集，所有线性运算符从$ x $到$ x $，我们可以考虑其仿射维度。我们建立了有关此维度可能值的几个结果。从$ x $和$ y $的尺寸方面，我们证明了最佳的上限。此外，我们在多面体规范空间中改善了这些估计值，用于给定维度的子空间的开放和致密子集。结果，在多面体规范的空间中，最小投影对于一个开放和致密的超平面子集是独特的。为了证明这一点，我们建立了Chalmers-Metcalf操作员的某些新属性。另一个结果是事实是，对于多面体规范空间的每个子空间，都有一个最小的投影，其中许多规范对。
Let $X$ be a finite-dimensional normed space and let $Y \subseteq X$ be its proper linear subspace. The set of all minimal projections from $X$ to $Y$ is a convex subset of the space all linear operators from $X$ to $X$ and we can consider its affine dimension. We establish several results on the possible values of this dimension. We prove optimal upper bounds in terms of the dimensions of $X$ and $Y$. Moreover, we improve these estimates in the polyhedral normed spaces for an open and dense subset of subspaces of the given dimension. As a consequence, in the polyhedral normed spaces a minimal projection is unique for an open and dense subset of hyperplanes. To prove this, we establish certain new properties of the Chalmers-Metcalf operator. Another consequence is the fact, that for every subspace of a polyhedral normed space, there exists a minimal projection with many norming pairs.
