In 1930, S. Warschawski [19] showed that H1(D), where D is the open unit disc in ℂ, has the following property: Let be a sequence of functions in H1(D) converging uniformly on compact subsets of D to a function f∈H1(D) and suppose that ‖fn‖1 = |f‖1 = 1 for all n∈ℕ. Then converges to zero. From a Banach space standpoint, this result says that H1(D) has the Kadec–Klee property with respect to uniform convergence on compact subsets of D. Warschawski's result was proved independently by Newman [16] in 1963 (see also [13] for another proof) and extended to more general domains by Hoffman [12], Goldstein and Swaminathan [8] and Godefroy [7]. A uniform version of Warschawski's result and its subsequent extensions was recently obtained by Besbes, Dilworth, Dowling and Lennard [2] (see also [1]). We mention here that these results for H1 spaces also hold for the Hp-spaces for 1 < p < ∞ because these spaces are uniformly convex.
UNIFORM CONVERGENCE OF MULTIPLIER CONVERGENT SERIES
