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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,[1] and, soon after, under the name of almost convergence, by Tadeusz Kuczumow.[2]
A sequence in a metric space is said to be Δ-convergent to if for every , .
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 (see [3]). In particular, Delta-convergence and weak convergence coincide if is a Hilbert space.
Coincidence of weak convergence and Delta-convergence is equivalent, for uniformly convex Banach spaces, to the well-known Opial property[3]
The Delta-compactness theorem of T. C. Lim[1] 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 (in the non-separable case) its proof does not depend on the Axiom of Choice.
An asymptotic center of a sequence , if it exists, is a limit of the Chebyshev centers for truncated sequences . A metric space is called asymptotically complete, if any bounded sequence in it has an asymptotic center.
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.[4]
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