Metric projection onto finite-dimensional subspaces of C and L

In the space C(Q) of real functions that are continuous on the compact set Q, a finite-dimensional subspace P will have a uniformly continuous metric projection if and only if Q is a finite sum of compact sets Qi, and either P is on each Qi a one-dimensional Chebyshev space, or x(t)≡0 for any x belonging to P. The metric projection onto any finite-dimensional subspace of the space La, b] of real integrable functions is not uniformly continuous.