A note in approximative compactness and midpoint locally k -uniform rotundity in Banach spaces

2018 ◽  
Vol 38 (2) ◽  
pp. 643-650
Author(s):  
Chunyan LIU ◽  
Zihou ZHANG ◽  
Yu ZHOU
Keyword(s):  
Banach Spaces ◽  
Uniform Rotundity ◽  
Approximative Compactness

Banach Spaces that are Uniformly Rotund in Weakly Compact Sets of Directions

1977 ◽  
Vol 29 (5) ◽  
pp. 963-970 ◽  
Author(s):  
Mark A. Smith
Keyword(s):  
Banach Space ◽  
Unit Ball ◽  
Banach Spaces ◽  
Nonzero Element ◽  
Bounded Subset ◽  
Compact Sets ◽  
Weakly Compact ◽  
Uniform Rotundity ◽  
Minimum Depth ◽  
Weakly Compact Sets

In a Banach space, the directional modulus of rotundity, δ (ϵ, z), measures the minimum depth at which the midpoints of all chords of the unit ball which are parallel to z and of length at least ϵ are buried beneath the surface. A Banach space is uniformly rotund in every direction (URED) if δ (ϵ, z) is positive for every positive ϵ and every nonzero element z. This concept of directionalized uniform rotundity was introduced by Garkavi [6] to characterize those Banach spaces in which every bounded subset has at most one Čebyšev center.

Approximative compactness and continuity of metric projector in Banach spaces and applications

2008 ◽  
Vol 51 (2) ◽  
pp. 293-303 ◽  
Author(s):  
ShuTao Chen ◽  
Henryk Hudzik ◽  
Wojciech Kowalewski ◽  
YuWen Wang ◽  
Marek Wisła
Keyword(s):  
Banach Spaces ◽  
Approximative Compactness

scholarly journals Characterization of approximative compactness in Banach spaces

2015 ◽  
Vol 45 (12) ◽  
pp. 1953-1960
Author(s):  
Yu ZHOU ◽  
ChunYan LIU ◽  
ZiHou ZHANG
Keyword(s):  
Banach Spaces ◽  
Approximative Compactness

scholarly journals On the weak uniform rotundity of Banach spaces

2003 ◽  
Vol 2003 (30) ◽  
pp. 1943-1945
Author(s):  
Wen D. Chang ◽  
Ping Chang
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Product Space ◽  
Quotient Space ◽  
Uniform Rotundity ◽  
Rotund Banach Space

We prove that ifXi,i=1,2,…,are Banach spaces that are weak* uniformly rotund, then theirlpproduct space(p>1)is weak* uniformly rotund, and for any weak or weak* uniformly rotund Banach space, its quotient space is also weak or weak* uniformly rotund, respectively.

Nearly dentability and approximative compactness and continuity of metric projector in Banach spaces

2011 ◽  
Vol 41 (9) ◽  
pp. 815-825 ◽  
Author(s):  
YunAn CUI ◽  
ShaoQiang SHANG ◽  
YongQiang FU
Keyword(s):  
Banach Spaces ◽  
Approximative Compactness

Some properties which generalize uniform rotundity of Banach spaces

1992 ◽  
Vol 59 (5) ◽  
pp. 457-467
Author(s):  
V. Montesinos ◽  
J. R. Torregrosa
Keyword(s):  
Banach Spaces ◽  
Uniform Rotundity

scholarly journals Approximative compactness and continuity of metric projections

1974 ◽  
Vol 11 (1) ◽  
pp. 47-55 ◽  
Author(s):  
B.B. Panda ◽  
O.P. Kapoor
Keyword(s):  
Large Class ◽  
Banach Spaces ◽  
Metric Space ◽  
Metric Projection ◽  
Uniformly Convex ◽  
Chebyshev Set ◽  
Approximatively Compact ◽  
Metric Projections ◽  
Uniformly Convex Spaces ◽  
Approximative Compactness

In the paper “Some remarks on approximative compactness”, Rev. Roumaine Math. Pures Appl. 9 (1964), Ivan Singer proved that if K is an approximatively compact Chebyshev set in a metric space, then the metric projection onto K is continuous. The object of this paper is to show that though, in general, the continuity of the metric projection supported by a Chebyshev set does not imply that the set is approximatively compact, it is indeed so in a large class of Banach spaces, including the locally uniformly convex spaces. It is also proved that in such a space X the metric projection onto a Chebyshev set is continuous on a set dense in X.

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