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[edit]

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

Characterization in Banach spaces[edit]

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

Opial property[edit]

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

Delta-compactness theorem[edit]

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

An asymptotic center of a sequence $(x_{k})_{k\in \mathbb {N} }$ , if it exists, is a limit of the Chebyshev centers $c_{n}$ for truncated sequences $(x_{k})_{k\geq n}$ . 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[edit]

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]

References[edit]

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

[Lim-1] 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.

[ST-3] 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.

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