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

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A sequence   in a metric space   is said to be Δ-convergent to   if for every  ,  .

Characterization in Banach spaces

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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.

Opial property

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

Delta-compactness theorem

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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.

Asymptotic center and asymptotic completeness

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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.

Uniform convexity as sufficient condition of asymptotic completeness

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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

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  • 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.

References

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  1. ^ a b T.C. Lim, Remarks on some fixed point theorems, Proc. Amer. Math. Soc. 60 (1976), 179–182.
  2. ^ T. Kuczumow, An almost convergence and its applications, Ann. Univ. Mariae Curie-Sklodowska Sect. A 32 (1978), 79–88.
  3. ^ a b S. Solimini, C. Tintarev, Concentration analysis in Banach spaces, Comm. Contemp. Math. 2015, DOI 10.1142/S0219199715500388
  4. ^ J. Staples, Fixed point theorems in uniformly rotund metric spaces, Bull. Austral. Math. Soc. 14 (1976), 181–192.