In mathematics, Delta-convergence, or Δ-convergence, is a mode of convergence in metric spaces, weaker than the usual metric convergence, and similar to (but distinct from) the weak convergence in Banach spaces. In Hilbert space, Delta-convergence and weak convergence coincide. For a general class of spaces, similarly to weak convergence, every bounded sequence has a Delta-convergent subsequence. Delta convergence was first introduced by Teck-Cheong Lim, and, soon after, under the name of almost convergence, by Tadeusz Kuczumow.
Contents
Definition
A sequence
Characterization in Banach spaces
If
Opial property
Coincidence of weak convergence and Delta-convergence is equivalent, for uniformly convex Banach spaces, to the well-known Opial property
Delta-compactness theorem
The Delta-compactness theorem of T. C. Lim states that if
The Delta-compactness theorem is similar to the Banach–Alaoglu theorem for weak convergence but, unlike the Banach-Alaoglu theorem (in the non-separable case) its proof does not depend on the Axiom of Choice.
Asymptotic center and asymptotic completeness
An asymptotic center of a sequence
Uniform convexity as sufficient condition of asymptotic completeness
Condition of asymptotic completeness in the Delta-compactness theorem is satisfied by uniformly convex Banach spaces, and more generally, by uniformly rotund metric spaces as defined by J. Staples.