Delta-convergence
In mathematics, Delta-convergence, or Δ-convergence, is a mode of convergence in metric spaces, weaker than the usual metric convergence, and similar to 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.
Definition
A sequence in a metric space is said to be Δ-convergent to if for every,.Characterization in Banach spaces
If is a uniformly convex and uniformly smooth Banach space, with the duality mapping given by,, then a sequence is Delta-convergent to if and only if converges to zero weakly in the dual space . In particular, Delta-convergence and weak convergence coincide if is a Hilbert space.Opial property
Coincidence of weak convergence and Delta-convergence is equivalent, for uniformly convex Banach spaces, to the well-knownOpial property
Delta-compactness theorem
The Delta-compactness theorem of T. C. Lim states that if is an asymptotically complete metric space, then every bounded sequence in has a Delta-convergent subsequence.The Delta-compactness theorem is similar to the Banach–Alaoglu theorem for weak convergence but, unlike the Banach-Alaoglu theorem its proof does not depend on the Axiom of Choice.