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 (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]

Definition

A sequence ${\displaystyle (x_{k})}$ in a metric space ${\displaystyle (X,d)}$ is said to be Δ-convergent to ${\displaystyle x\in X}$ if for every ${\displaystyle y\in X}$, ${\displaystyle \limsup(d(x_{k},x)-d(x_{k},y))\leq 0}$.

Characterization in Banach spaces

If ${\displaystyle X}$ is a uniformly convex and uniformly smooth Banach space, with the duality mapping ${\displaystyle x\mapsto x^{*}}$ given by ${\displaystyle \|x\|=\|x^{*}\|}$, ${\displaystyle \langle x^{*},x\rangle =\|x\|^{2}}$, then a sequence ${\displaystyle (x_{k})\subset X}$ is Delta-convergent to ${\displaystyle x}$ if and only if ${\displaystyle (x_{k}-x)^{*}}$ converges to zero weakly in the dual space ${\displaystyle X^{*}}$ (see ^[3]). In particular, Delta-convergence and weak convergence coincide if ${\displaystyle X}$ is a Hilbert space.

Opial property

Coincidence of weak convergence and Delta-convergence is equivalent, for uniformly convex Banach spaces, to the well-known Opial property^[3]

Delta-compactness theorem

The Delta-compactness theorem of T. C. Lim^[1] states that if ${\displaystyle (X,d)}$ is an asymptotically complete metric space, then every bounded sequence in ${\displaystyle X}$ 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.

Asymptotic center and asymptotic completeness

An asymptotic center of a sequence ${\displaystyle (x_{k})_{k\in \mathbb {N} }}$, if it exists, is a limit of the Chebyshev centers ${\displaystyle c_{n}}$ for truncated sequences ${\displaystyle (x_{k})_{k\geq n}}$. A metric space is called asymptotically complete, if any bounded sequence in it has an asymptotic center.

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.^[4]

Further reading

• William Kirk, Naseer Shahzad, Fixed point theory in distance spaces. Springer, Cham, 2014. xii+173 pp.
• G. Devillanova, S. Solimini, C. Tintarev, On weak convergence in metric spaces, Nonlinear Analysis and Optimization (B. S. Mordukhovich, S. Reich, A. J. Zaslavski, Editors), 43–64, Contemporary Mathematics 659, AMS, Providence, RI, 2016.