Metric projection onto finite-dimensional subspaces of $\mathrm{C}$ and $\mathrm{L}$

V. I. Berdyshev

Matematičeskie zametki, Tome 18 (1975) no. 4, pp. 473-488.

Résumé

In the space $\mathrm{C(Q)}$ of real functions that are continuous on the compact set $\mathrm{Q}$, a finite-dimensional subspace $\mathrm{P}$ will have a uniformly continuous metric projection if and only if $\mathrm{Q}$ is a finite sum of compact sets $\mathrm{Q_i}$, and either $\mathrm{P}$ is on each $\mathrm{Q_i}$ a one-dimensional Chebyshev space, or $\mathrm{x(t)\equiv0\mathrm}$ for any $\mathrm{x}$ belonging to $\mathrm{P}$. The metric projection onto any finite-dimensional subspace of the space $\mathrm{L[a, b]}$ of real integrable functions is not uniformly continuous.