On the existence question for a family of products

Gary Allen Gislason

Vol. 34 (1970), No. 2, 385–388

DOI: 10.2140/pjm.1970.34.385

Abstract

Let X be a topological space and let P and Q be finite dimensional linear subspaces of C(X). Since the set PQ = {pq : p ∈ P,q ∈ Q} is a subset of a finite dimensional linear subspace of C(X), existence of best approximations from PQ is assured if and only if PQ is closed. If p ∈ P,q ∈ Q, and pq = 0 imply that p = 0 or q = 0, then PQ is shown to be closed. An example shows that PQ is not closed in general.

Mathematical Subject Classification

Primary: 41.60

Secondary: 46.00

Milestones

Received: 30 October 1969

Published: 1 August 1970