Geodesic
Banach Spaces with the PC Property
Matematičeskie zametki, Tome 76 (2004) no. 4, pp. 568-577.

Voir la notice de l'article provenant de la source Math-Net.Ru

A Banach space $X$ possesses the PC (point of continuity) property if for any $w$-closed bounded subset $A\subset X$ the identity map $(A,w)\to(A,\|\cdot\|)$ has a point of continuity ($w$ is the weak topology in $X$). We deduce some criteria for Banach spaces to have the PC property and describe (for dual Banach spaces) relationships between spaces possessing the PC property and spaces possessing the RN or the WRN property. 
@article{MZM_2004_76_4_a8,
     author = {V. I. Rybakov},
     title = {Banach {Spaces} with the {PC} {Property}},
     journal = {Matemati\v{c}eskie zametki},
     pages = {568--577},
     publisher = {mathdoc},
     volume = {76},
     number = {4},
     year = {2004},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2004_76_4_a8/}
}
TY  - JOUR
AU  - V. I. Rybakov
TI  - Banach Spaces with the PC Property
JO  - Matematičeskie zametki
PY  - 2004
SP  - 568
EP  - 577
VL  - 76
IS  - 4
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/MZM_2004_76_4_a8/
LA  - ru
ID  - MZM_2004_76_4_a8
ER  - 
%0 Journal Article
%A V. I. Rybakov
%T Banach Spaces with the PC Property
%J Matematičeskie zametki
%D 2004
%P 568-577
%V 76
%N 4
%I mathdoc
%U http://geodesic.mathdoc.fr/item/MZM_2004_76_4_a8/
%G ru
%F MZM_2004_76_4_a8
V. I. Rybakov. Banach Spaces with the PC Property. Matematičeskie zametki, Tome 76 (2004) no. 4, pp. 568-577. http://geodesic.mathdoc.fr/item/MZM_2004_76_4_a8/

[1] Edgar G. A., Wheeller R. F., “Topological properties of Banach spaces”, Pacific J. Math., 115:2 (1984), 317–350 | MR | Zbl

[2] Arkhangelskii A. A., “Kompaktnost”, Itogi nauki i tekhniki. Sovremennye problemy matematiki. Fundamentalnye napravleniya, 50, VINITI, M., 1989, 5–128

[3] Namioka I., Pol R., “$\sigma$-fragmentability and analyticity”, Mathematika, 43 (1996), 172–181 | MR | Zbl

[4] Saint-Raymond J., “Jeux topologiques et espaces de Namioka”, Proc. Amer. Math. Soc., 87:3 (1983), 499–504 | DOI | MR | Zbl

[5] Engelking R., Obschaya topologiya, Mir, M., 1986

[6] Mercourakis S., Negrepontis S., “Banach spaces and topology, II”, Recent Progress in General Topology (Prague, 1991), North-Holland, Amsterdam, 1992, 493–536 | MR | Zbl

[7] Deville R., Godefroy G., Zizler V., Smoothness and Renormings in Banach Spaces, Pitman, New York, 1993 | Zbl

[8] Fonf V. P., “O banakhovykh prostranstvakh, obladayuschikh svoistvom (PC)”, Dokl. Bolgarskoi AN, 43:5 (1990), 15–16 | MR | Zbl

[9] Stegall C., “Functions of the first Baire class with values in Banach spaces”, Proc. Amer. Math. Soc., 111:4 (1991), 481–491 | DOI | MR

[10] Burbaki N., Obschaya topologiya, Nauka, M., 1975

[11] Bourgin R. D., Geometric Aspects of Convex Sets with Radon–Nikodým property, Lecture Notes in Math., 993, Springer-Verlag, New York, 1983 | MR | Zbl

[12] Ridde L., Uhl J. J., “Martingales and the fine line between Asplund spaces and spaces not containing a copy of $\ell_1$”, Lecture Notes in Math., 939, Springer-Verlag, New York, 1981, 145–156

[13] Musial K., “Topics in the theory of Pettis integration”, Rend. Inst. Mat. Univ. Trieste, 23 (1991), 177–262 | MR | Zbl

[14] Ghoussoub N., Maurey B., “$G_\delta$-embeddings in Hilbert space”, J. Funct. Anal., 61 (1985), 72–97 | DOI | MR | Zbl

[15] Rosenthal H., “A characterization of Banach spaces containing $\ell_1$”, Proc. Nat. Acad. Sci. USA, 71 (1971), 2411–2413 | DOI

[16] Lindenstrauss Y., Stegall C., “Examples of separable spaces which do not contain $\ell_1$ and whose duals are not separable”, Studia Math., 54 (1975), 81–105 | MR | Zbl

[17] Ghoussoub N., Maurey B., Schachermayer W., “A counterexample to a problem on points of continuity in Banach spaces”, Proc. Amer. Math. Soc., 92 (1987), 278–282 | DOI | MR

[18] Diestel J., Uhl J. J., Vector Measures, Math. Surveys, 15, Amer. Math. Soc., Providence, RI, 1977 | MR | Zbl