×

Rough convergence in infinite dimensional normed spaces. (English) Zbl 1029.46005

Summary: For given \(r\), \(\rho\geq 0\), a sequence \((x_i)\) in some normed linear space \(X\) is said to be \(r\)-convergent if the \(r\)-limit set defined by \[ \text{LIM}^r x_i=\bigl\{x_*\in X:\limsup_{i\to\infty} \|x_i-x_* \|\leq r\bigr\} \] is nonempty, and it is called a \(\rho\)-Cauchy sequence if \[ \forall \varepsilon >0\;\exists i_\varepsilon:i,j\geq i_\varepsilon \Rightarrow \|x_i-x_j \|< \rho+ \varepsilon. \] This article investigates different aspects of this rough convergence, especially in infinite dimensional spaces, such as analytical properties of the limit set, relation to other convergence notions, and the dependence of the \(r\)-limit set on the roughness degree \(r\). Moreover, by using the Jung constant we find the minimal value of \(r\) such that an arbitrary \(\rho\)-Cauchy sequence in \(X\) is certainly \(r\)-convergent.

MSC:

46B03 Isomorphic theory (including renorming) of Banach spaces
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Amir D., Pacific J. Math. 118 pp 1– (1985) · Zbl 0529.46011 · doi:10.2140/pjm.1985.118.1
[2] Appell J., Israel J. Math. 116 pp 171– (2000) · Zbl 0977.46004 · doi:10.1007/BF02773217
[3] Aubin J.-P., Set-Valued Analysis (1990)
[4] Ball K., Israel J. Math. 58 pp 243– (1987) · Zbl 0642.46021 · doi:10.1007/BF02785681
[5] Bohnenblust F., Ann. of Math. 39 pp 301– (1938) · Zbl 0019.14101 · doi:10.2307/1968786
[6] Bouligand G., Ens. Math. 31 pp 14– (1932)
[7] Davis W., J. Approx. Theory 21 pp 315– (1977) · Zbl 0373.46034 · doi:10.1016/0021-9045(77)90001-6
[8] Grünbaum B., Pacific J. Math. 9 pp 487– (1959) · Zbl 0086.15203 · doi:10.2140/pjm.1959.9.487
[9] Jung H. E. W., J. Reine Angew. Math. 123 pp 241– (1901)
[10] Kuratowski K., Fund. Math. 18 pp 148– (1932)
[11] Leichtweiss K., Math. Z. 62 pp 37– (1955) · Zbl 0064.16701 · doi:10.1007/BF01180623
[12] Leichtweiss K., Konvexe Mengen (1980) · doi:10.1007/978-3-642-95335-4
[13] Maluta E., Pacific J. Math. 111 pp 357– (1984) · Zbl 0495.46012 · doi:10.2140/pjm.1984.111.357
[14] Phu H. X., Numer. Funct. Anal. Optim. 22 pp 201– (2001)
[15] Pichugov S. A., Mat. Zemetki 43 pp 604– (1988)
[16] Routledge N., Quart. J. Math. 3 pp 12– (1952) · Zbl 0046.12301 · doi:10.1093/qmath/3.1.12
[17] Semenov E. M., St. Petersburg Math. J. 10 pp 861– (1999)
[18] Zeidler E., Nonlinear Functional Analysis and Its Applications I (1986) · Zbl 0583.47050 · doi:10.1007/978-1-4612-4838-5
[19] Zeidler E., Nonlinear Functional Analysis and Its Applications II (1990) · Zbl 0684.47029
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.