Spaces of continuous functions
Sumner Byron Myers · Bulletin of the American Mathematical Society · 1949
Let X be a completely regular topological space, B(X) the Banach space of real-valued bounded continuous functions on X, with the usual norm ||&|| =sup a? £x|&(#)| • A subset GCB(X) is called completely regular (c.r.) over X if given any closed subset KQ.X and point XoÇzX -K, there exists a ô£G such that &(#o) = |NI an( i sup^^is: \b(x)\ sup^^ b(x) <||&||, then a closed linear subspace of B(X) c.r. over X does not necessarily determine the topology of X.For example, if X consists of just two points, xi and X2, then the subspace G of B(X) consisting of all bÇ:B(X) such that b(xi) --b(x 2 ) is c.r. over X according to the weakened definition, yet it is equivalent to the space B(Y), where Y consists of a single point.Proper closed linear subspaces of B{X) which are c.r. over X exist in general for both compact and non-compact X, and may contain the constant functions.This is in contrast to the situation when B{X) is made into a normed ring (Banach algebra) R(X) or into a Banach lattice L(X); if X is compact, no proper closed subring of R(X) containing the constant functions can be c.r. over X [s] t and no proper closed sublattice of L(X) containing the constant functions can be c.r. over X [4].Since topological properties of X must be reflected in metric and Presented to the Society, February 28, 1948; received by the editors April 5, 1948. 1 Numbers in brackets refer to the bibliography at the end of the paper.2 "Compact" means bicompact and Hausdorff.