A new class of weakly K -analytic Banach spaces

Sophocles Mercourakis; E. Stamati

Commentationes Mathematicae Universitatis Carolinae (2006)

  • Volume: 47, Issue: 2, page 291-312
  • ISSN: 0010-2628

Abstract

top
In this paper we define and investigate a new subclass of those Banach spaces which are K -analytic in their weak topology; we call them strongly weakly K -analytic (SWKA) Banach spaces. The class of SWKA Banach spaces extends the known class of strongly weakly compactly generated (SWCG) Banach spaces (and their subspaces) and it is related to that in the same way as the familiar classes of weakly K -analytic (WKA) and weakly compactly generated (WCG) Banach spaces are related. We show that: (i) not every separable Banach space is SWKA; (ii) every separable SWKA Banach space not containing ℓ 1 is Polish; (iii) we answer in the negative a question posed in [S-W] by constructing a subspace X of the SWCG space L 1 [ 0 , 1 ] which is not SWCG.

How to cite

top

Mercourakis, Sophocles, and Stamati, E.. "A new class of weakly $K$-analytic Banach spaces." Commentationes Mathematicae Universitatis Carolinae 47.2 (2006): 291-312. <http://eudml.org/doc/249885>.

@article{Mercourakis2006,
abstract = {In this paper we define and investigate a new subclass of those Banach spaces which are $K$-analytic in their weak topology; we call them strongly weakly $K$-analytic (SWKA) Banach spaces. The class of SWKA Banach spaces extends the known class of strongly weakly compactly generated (SWCG) Banach spaces (and their subspaces) and it is related to that in the same way as the familiar classes of weakly $K$-analytic (WKA) and weakly compactly generated (WCG) Banach spaces are related. We show that: (i) not every separable Banach space is SWKA; (ii) every separable SWKA Banach space not containing $\ell ^1$ is Polish; (iii) we answer in the negative a question posed in [S-W] by constructing a subspace $X$ of the SWCG space $L^1[0,1]$ which is not SWCG.},
author = {Mercourakis, Sophocles, Stamati, E.},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {WKA; SWKA Banach spaces; $K$-analytic space; Baire space; Polish space; WKA; SWKA Banach spaces; -analytic space; Baire space; Polish space},
language = {eng},
number = {2},
pages = {291-312},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {A new class of weakly $K$-analytic Banach spaces},
url = {http://eudml.org/doc/249885},
volume = {47},
year = {2006},
}

TY - JOUR
AU - Mercourakis, Sophocles
AU - Stamati, E.
TI - A new class of weakly $K$-analytic Banach spaces
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2006
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 47
IS - 2
SP - 291
EP - 312
AB - In this paper we define and investigate a new subclass of those Banach spaces which are $K$-analytic in their weak topology; we call them strongly weakly $K$-analytic (SWKA) Banach spaces. The class of SWKA Banach spaces extends the known class of strongly weakly compactly generated (SWCG) Banach spaces (and their subspaces) and it is related to that in the same way as the familiar classes of weakly $K$-analytic (WKA) and weakly compactly generated (WCG) Banach spaces are related. We show that: (i) not every separable Banach space is SWKA; (ii) every separable SWKA Banach space not containing $\ell ^1$ is Polish; (iii) we answer in the negative a question posed in [S-W] by constructing a subspace $X$ of the SWCG space $L^1[0,1]$ which is not SWCG.
LA - eng
KW - WKA; SWKA Banach spaces; $K$-analytic space; Baire space; Polish space; WKA; SWKA Banach spaces; -analytic space; Baire space; Polish space
UR - http://eudml.org/doc/249885
ER -

References

top
  1. Argyros S., Mercourakis S., On weakly Lindelöf Banach spaces, Rocky Mountain J. Math. 23 395-446 (1993). (1993) Zbl0797.46009MR1226181
  2. Christensen J.P.R., Topology and Borel Structure, North-Holland, Amsterdam, 1974. Zbl0273.28001MR0348724
  3. Deville R., Godefroy G., Zizler V., Smoothness and renorming in Banach spaces, Longman, Harlow, 1993. MR1211634
  4. Engelking R., General Topology, PWN, Warszawa, 1977. Zbl0684.54001MR0500780
  5. Edgar G.A., Wheller R.F., Topological properties of Banach spaces, Pacific J. Math. 115 2 317-350 (1984). (1984) MR0765190
  6. Habala P., Hájek P., Zizler V., Introduction to Banach Spaces I and II, Matfyzpress, Praha, 1996. 
  7. Jayne J.E., Rogers C.A., K -analytic sets, in Analytic Sets, Academic Press, London, 1980. Zbl0589.54047
  8. Kuratowski K., Topology, Vol. I (1966), Vol. II (1968), Academic Press, New York-London. Zbl0849.01044MR0217751
  9. Lindenstrauss J., Tzafriri L., Classical Banach Spaces I, Springer, Berlin-New York, 1977. Zbl0362.46013MR0500056
  10. Mercourakis S., On weakly countably determined Banach spaces, Trans. Amer. Math. Soc. 300 307-327 (1987). (1987) Zbl0621.46018MR0871678
  11. Mercourakis S., Negrepontis S., Banach Spaces and Topology II, Recent Progress in General Topology, M. Hušek and J. Van Mill (eds.), North-Holland, Amsterdam, 1992, pp.493-536. Zbl0832.46005MR1229137
  12. Negrepontis S., Banach Spaces and Topology, Handbook of Set-Theoretic Topology, K. Kunen and J. Vaughan (eds.), North-Holland, Amsterdam, 1984, pp.1045-1142. Zbl0832.46005MR0776642
  13. Rosenthal H.P., A characterization of Banach spaces containing ℓ 1 , Proc. Nat. Acad. Sci. (USA) 71 2411-2413 (1974). (1974) MR0358307
  14. Rosenthal H.P., The heredity problem for weakly compactly generated Banach spaces, Compositio Math. 28 83-111 (1974). (1974) Zbl0298.46013MR0417762
  15. Rosenthal H.P., Weak*-Polish Banach spaces, J. Funct. Anal. 76 267-316 (1988). (1988) Zbl0655.46011MR0924462
  16. Stegall C., The Radon-Nikodym property in conjugate Banach spaces, Trans. Amer. Math. Soc. 206 213-223 (1975). (1975) Zbl0318.46056MR0374381
  17. Stern J., A Ramsey theorem for trees, with an application to Banach spaces, Israel J. Math. 29 179-188 (1978). (1978) Zbl0378.46012MR0476554
  18. Schlüchtermann G., Wheeler R.F., On Strongly WCG Banach spaces, Math. Z. 199 387-398 (1988). (1988) MR0961818
  19. Talagrand M., Espaces de Banach faiblement 𝒦 -analytiques, Ann. of Math. 110 407-438 (1979). (1979) MR0554378
  20. Todorčević S., Compact subsets of the first Baire class, J. Amer. Math. Soc. 12 4 1179-1212 (1999). (1999) MR1685782

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.