Conditional weak compactness in vector-valued function spaces

Author:
Marian Nowak

Journal:
Proc. Amer. Math. Soc. **129** (2001), 2947-2953

MSC (2000):
Primary 46B25, 46E40

DOI:
https://doi.org/10.1090/S0002-9939-01-06064-6

Published electronically:
April 17, 2001

MathSciNet review:
1840098

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Let be an ideal of over a -finite measure space and let be the Köthe dual of with . Let be a real Banach space, and the topological dual of . Let be a subspace of the space of equivalence classes of strongly measurable functions and consisting of all those for which the scalar function belongs to . For a subset of for which the set is -bounded the following statement is equivalent to conditional -compactness: the set is conditionally -compact and is a conditionally weakly compact subset of for each , with . Applications to Orlicz-Bochner spaces are given.

**[AB]**C.D. Aliprantis and O. Burkinshaw,*Locally solid Riesz spaces*, Academic Press, New York, San Francisco, London, 1978. MR**58:12271****[ABBL]**C. Abbott, E. Bator, R. Bilyeu, P. Lewis,*Weak precompactness, strong boundedness, and weak complete continuity*, Math. Proc. Camb. Phil. Soc.,**108**(1990), 325-335. MR**92b:46047****[BH]**J. Batt, W. Hiermeyer,*On compactness in in the weak topology and in the topology*, Math. Z.,**182**(1983), 409-423. MR**84m:46039****[B]**F. Bombal,*On the space*, Rev. Real Acad. Cienc. Exact. Fis. Natur. Madrid,**74**, no.**1**(1980), 131-135 (in Spanish). MR**82i:46042****[B]**F. Bombal,*On Orlicz space of vector-valued functions*, Collect. Math.,**32**, no.**1**(1981), 3-12 (in Spanish). MR**83a:46039****[BL]**R. Bilyeu, P. Lewis,*Uniform differentiability, uniform absolute continuity and the Vitali-Hahn-Saks theorem*, Rocky Mtn. J. Math.,**10**, No. 3 (1980), 533-557. MR**82g:46083****[Bu]**A.V. Bukhvalov,*On an analytic representation of operators with abstract norm*, Izv. Vyss. Uceb. Zaved.,**11**(1975), 21-32 (in Russian).**[DU]**J. Diestel, J.J. Uhl Jr.,*Vector measures*, Math. Surveys,**15**, Amer. Math. Soc., Providence, R.I., 1977. MR**56:12216****[D]**J. Diestel,*Sequences and series in Banach spaces*, Graduate Texts in Math., Springer-Verlag, New York, 1984. MR**85i:46020****[KA]**L.V. Kantorovitch, A.V. Akilov,*Functional analysis*, Nauka, Moscow, 1984 (3 ed.) (in Russian).**[KR]**M. Krasnoselskii, Ya. B. Rutickii,*Convex functions and Orlicz spaces*, P. Noordhoff Ltd, Groningen, 1961. MR**23:A4016****[L]**W. Luxemburg,*Banach function spaces*, Delft, 1955. MR**17:285a****[MN]**P. Mayer-Nieberg,*Banach lattices*, Springer-Verlag, Berlin, Heidelberg, New York, 1991.**[N]**M. Nowak,*Order continuous seminorms and weak compactness in Orlicz spaces*, Collect. Math.,**44**(1993), 217-236. MR**95g:46055****[N]**M. Nowak,*Weak sequential compactness in non-locally convex Orlicz spaces*, Bull. Pol. Acad. Sci.,**46**(1998), 225-231. MR**99g:46034****[R]**H.P. Rosenthal,*A characterization of Banach spaces containing*, Proc. Nat. Acad. Sci. U.S.A.,**71**(1974), 2411-2413. MR**50:10773****[Z]**A.C. Zaanem,*Riesz spaces*II, North Holland Pub. Comp., Amsterdam, New York, Oxford, 1983.

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (2000):
46B25,
46E40

Retrieve articles in all journals with MSC (2000): 46B25, 46E40

Additional Information

**Marian Nowak**

Affiliation:
Institute of Mathematics, T. Kotarbiński Pedagogical University, Pl. Słowiański 9, 65–069 Zielona Góra, Poland

Email:
mnowa@lord.wsp.zgora.pl

DOI:
https://doi.org/10.1090/S0002-9939-01-06064-6

Keywords:
Conditional weak compactness,
vector valued function spaces

Received by editor(s):
July 6, 1998

Received by editor(s) in revised form:
February 14, 2000

Published electronically:
April 17, 2001

Communicated by:
Dale Alspach

Article copyright:
© Copyright 2001
American Mathematical Society