Showing papers on "Dual norm published in 1974"

Order and norm convergence in Banach lattices

Abstract: Let(V, ≧, ‖ · ‖) be a Banach lattice, and denote V\{0} by V'. For the definition of a Banach lattice and other undefined terms used below, see Vulikh [4]. Leader [3] shows that, if norm convergence is equivalent to order convergence for sequences in V, then the norm is equivalent to an M-norm. By assuming the equivalence for nets in V we can strengthen this result.

Variation norm convergence of function sequences

Abstract: We prove that a pointwise convergent sequence of convex functions with a continuous limit converges with respect to the total variation norm. This yields a theorem on convexity-preserving operators which has as a corollary the result that a complex function / is absolutely continuous on [O, l] if and only if the sequence B.(f) of Bernstein polynomials of / converges to / with respect to the total variation norm. In this paper a theorem which is analogous to Dini's theorem is proved; Theorem 1. // f. is a pointwise convergent sequence of real-valued functions, each of which is convex on [a, b] and the limit function F is continuous on [a, b\, then the sequence fm converges to F with respect to the total variation norm on [a, b\. This is then used to prove Theorem 2. Suppose T. is a sequence of linear operators from AC[a, b] into AC[a, b] such that for each f e AC\a, b], (1) T.(f) converges pointwise to f on [a, b\; (2) if f is convex on [a, b] and n is a nonnegative integer, T (/) z's convex on [a, b\; and (3) there is a number Al > 0 such that for each nonnegative integer n, y\a\T (f))\ < Mf^\df\. Then, for each f e AC[a, b], the function sequence T'.(/) converges to f with respect to the total variation norm. Corollary. A complex-valued function f is absolutely continuous on Presented to the Society, January 25, 1973; received by the editors March 10, 1973 and, in revised form, October 22, 197 3. AMS (MOS) subject classifications (1970). Primary 41A30, 41A25; Secondary 26A51.