Weakly compact sets in H1

Suppose that A is a uniform algebra on a compact set X and that ϕ: A → C is a nonzero multiplicative linear functional on A. Let Mϕ be the set of positive representing measures for ϕ. If Mϕ is finite dimensional, let m be a core measure of Mϕ. The space H1 is the closure of A in L1m). The space H∞ is the weak* (i.e. σ(L̞L1)) closure of A in L̞(m). The weakly compact sets R in H1 are then those sets such that for all ∈ > 0 there is a bounded set in H∞ which approximates R up to ∈. © 1976 by Pacific Journal of Mathematics.SCOPUS: ar.j