Topological dual of non-locally convex Orlicz-Bochner spaces

Marian Nowak

Commentationes Mathematicae Universitatis Carolinae (1999)

  • Volume: 40, Issue: 3, page 511-529
  • ISSN: 0010-2628

Abstract

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Let L ϕ ( X ) be an Orlicz-Bochner space defined by an Orlicz function ϕ taking only finite values (not necessarily convex) over a σ -finite atomless measure space. It is proved that the topological dual L ϕ ( X ) * of L ϕ ( X ) can be represented in the form: L ϕ ( X ) * = L ϕ ( X ) n ∼ ⊕ L ϕ ( X ) s ∼ , where L ϕ ( X ) n ∼ and L ϕ ( X ) s ∼ denote the order continuous dual and the singular dual of L ϕ ( X ) respectively. The spaces L ϕ ( X ) * , L ϕ ( X ) n ∼ and L ϕ ( X ) s ∼ are examined by means of the H. Nakano’s theory of conjugate modulars. (Studia Mathematica 31 (1968), 439–449). The well known results of the duality theory of Orlicz spaces are extended to the vector-valued setting.

How to cite

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Nowak, Marian. "Topological dual of non-locally convex Orlicz-Bochner spaces." Commentationes Mathematicae Universitatis Carolinae 40.3 (1999): 511-529. <http://eudml.org/doc/248418>.

@article{Nowak1999,
abstract = {Let $L^\varphi (X)$ be an Orlicz-Bochner space defined by an Orlicz function $\varphi $ taking only finite values (not necessarily convex) over a $\sigma $-finite atomless measure space. It is proved that the topological dual $L^\varphi (X)^*$ of $L^\varphi (X)$ can be represented in the form: $L^\varphi (X)^*=L^\varphi (X)^\sim _n\oplus L^\varphi (X)^\sim _s$, where $L^\varphi (X)^\sim _n$ and $L^\varphi (X)^\sim _s$ denote the order continuous dual and the singular dual of $L^\varphi (X)$ respectively. The spaces $L^\varphi (X)^*$, $L^\varphi (X)^\sim _n$ and $L^\varphi (X)^\sim _s$ are examined by means of the H. Nakano’s theory of conjugate modulars. (Studia Mathematica 31 (1968), 439–449). The well known results of the duality theory of Orlicz spaces are extended to the vector-valued setting.},
author = {Nowak, Marian},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {vector-valued function spaces; Orlicz functions; Orlicz spaces; Orlicz-Bochner spaces; topological dual; order dual; order continuous linear functionals; singular linear functionals; modulars; conjugate modulars; Orlicz function; Orlicz spaces; Orlicz-Bochner spaces; topological dual; order continuous dual; singular dual},
language = {eng},
number = {3},
pages = {511-529},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Topological dual of non-locally convex Orlicz-Bochner spaces},
url = {http://eudml.org/doc/248418},
volume = {40},
year = {1999},
}

TY - JOUR
AU - Nowak, Marian
TI - Topological dual of non-locally convex Orlicz-Bochner spaces
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 1999
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 40
IS - 3
SP - 511
EP - 529
AB - Let $L^\varphi (X)$ be an Orlicz-Bochner space defined by an Orlicz function $\varphi $ taking only finite values (not necessarily convex) over a $\sigma $-finite atomless measure space. It is proved that the topological dual $L^\varphi (X)^*$ of $L^\varphi (X)$ can be represented in the form: $L^\varphi (X)^*=L^\varphi (X)^\sim _n\oplus L^\varphi (X)^\sim _s$, where $L^\varphi (X)^\sim _n$ and $L^\varphi (X)^\sim _s$ denote the order continuous dual and the singular dual of $L^\varphi (X)$ respectively. The spaces $L^\varphi (X)^*$, $L^\varphi (X)^\sim _n$ and $L^\varphi (X)^\sim _s$ are examined by means of the H. Nakano’s theory of conjugate modulars. (Studia Mathematica 31 (1968), 439–449). The well known results of the duality theory of Orlicz spaces are extended to the vector-valued setting.
LA - eng
KW - vector-valued function spaces; Orlicz functions; Orlicz spaces; Orlicz-Bochner spaces; topological dual; order dual; order continuous linear functionals; singular linear functionals; modulars; conjugate modulars; Orlicz function; Orlicz spaces; Orlicz-Bochner spaces; topological dual; order continuous dual; singular dual
UR - http://eudml.org/doc/248418
ER -

References

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  1. Aliprantis C.D., Burkinshaw O., Locally solid Riesz spaces, Academic Press, New York, 1978. Zbl1043.46003MR0493242
  2. Ando T., Linear functionals on Orlicz spaces, Niew Arch. Wisk. 8 (1960), 1-16. (1960) Zbl0129.08001MR0123907
  3. Behrends E., M-structure and the Banach-Stone theorem, Lecture Notes in Math. 736, Springer-Verlag, Berlin-Heidelberg-New York, 1979. Zbl0371.46013MR0547509
  4. Bochner S., Taylor A.E., Linear functionals on certain spaces of abstractly-valued functions, Ann. of Math. (2) 39 (1938), 913-944. (1938) Zbl0020.37101MR1503445
  5. Bukhvalov A.V., On an analytic representation of operators with abstract norm, Soviet Math. Dokl. 14 (1973), 197-201. (1973) Zbl0283.47028
  6. Bukhvalov A.V., On an analytic representation of operators with abstract norm (in Russian), Izv. Vyss. Uceb. Zaved. 11 (1975), 21-32. (1975) MR0470746
  7. Bukhvalov A.V., On an analytic representation of linear operators by vector-valued measurable functions (in Russian), Izv. Vyss. Uceb. Zaved. 7 (1977), 21-31. (1977) 
  8. Bukhvalov A.V., Lozanowskii G.Ya., On sets closed in measure in spaces of measurable functions, Trans. Moscow Math. Soc. 2 (1978), 127-148. (1978) 
  9. Chen S., Hudzik H., On some convexities of Orlicz and Orlicz-Bochner spaces, Comment. Math. Univ. Carolinae 29.1 (1988), 13-29. (1988) Zbl0647.46030MR0937545
  10. Day M.M., The spaces L p with 0 < p < 1 , Bull. Amer. Math. Soc. 46 (1940), 816-823. (1940) MR0002700
  11. Drewnowski L., Compact operators on Musielak-Orlicz spaces, Comment. Math. 27 (1988), 225-232. (1988) Zbl0676.46024MR0978274
  12. Escribano C., Fernández F.J., Jiminéz Guerra P., On the Orlicz spaces of functions valued in locally convex spaces, Comment. Math. 32 (1992), 33-44. (1992) MR1202756
  13. Gramsch B., Die Klasse metrischer linear Raüme ℒ Φ , Math. Ann. 176 (1967), 61-78. (1967) MR0209819
  14. Hernandez F., Continuous functionals on a class of Orlicz spaces, Houston J. Math. 11 2 (1985), 171-181. (1985) Zbl0577.46025MR0792193
  15. Ionescu Tulcea A., Ionescu Tulcea C., Topics in the Theory of Lifting, Springer-Verlag, Berlin-Heidelberg-New York, 1969. Zbl0179.46303MR0276438
  16. Kamińska A., Turett B., Uniformly non- l 1 ( n ) Orlicz-Bochner spaces, Bull. Polon. Acad. Sci. 35 3-4 (1987), 211-218. (1987) MR0908170
  17. Kantorovich L.V., Akilov G.P., Functional Analysis (in Russian), Nauka, Moscow, 1984 (3 edition). MR0788496
  18. Luxemburg W.A., Banach Function Spaces, Delft, 1955. Zbl0162.44701MR0072440
  19. Luxemburg W.A., Zaanen A.C., Conjugate spaces of Orlicz spaces, Indagationes Math. 59 (1956), 217-228. (1956) Zbl0072.12301MR0077899
  20. Maligranda L., Wnuk W., Landau type theorem for Orlicz spaces, Math. Z. 208 (1991), 57-64. (1991) Zbl0738.46013MR1125732
  21. Musielak J., Orlicz W., Some remarks on modular spaces, Bull. Acad. Polon. Sci. 7 (1959), 661-668. (1959) Zbl0099.09202MR0112017
  22. Musielak J., Orlicz spaces and modular spaces, Lecture Notes Math. 1034, Springer-Verlag, Berlin, Heidelberg, New York, Tokyo, 1983. Zbl0557.46020MR0724434
  23. Nakano H., On generalized modular spaces, Studia Math. 31 (1968), 439-449. (1968) MR0234248
  24. Nowak M., On the order structure of Orlicz lattices, Bull. Acad. Polon. Sci. 36 5-6 (1988), 239-249. (1988) Zbl0767.46005MR1101668
  25. Nowak M., Order continuous linear functionals on non-locally convex Orlicz spaces, Comment. Math. Univ. Carolinae 33.3 (1992), 465-475. (1992) Zbl0812.46017MR1209288
  26. Nowak M., Duality of non-locally convex Orlicz spaces, Math. Japonica 5 (1993), 813-832. (1993) Zbl0801.46029MR1240281
  27. Nowak M., Duality theory of vector-valued function spaces I, Comment. Math. 37 (1997), 195-215. (1997) Zbl0908.46023MR1608189
  28. Nowak M., A note on generalized Young's inequality for Orlicz functions, Funct. et Approx. 26 (1998), 3 225-231. (1998) Zbl0927.46020MR1666621
  29. Nowak M., Mackey topologies of Orlicz-Bochner spaces, Proc. Razmadze Math. Inst. 118 (1998), 101-110. (1998) Zbl0951.46011MR1693247
  30. Orlicz W., A note on modular spaces I, Bull. Acad. Polon. Sci. 9 (1961), 157-162. (1961) Zbl0109.33404MR0131763
  31. Orlicz W., On integral representability of linear functionals over the space of ϕ -integrable functions, Bull. Acad. Polon. Sci. 8 (1960), 567-569. (1960) MR0126160
  32. Rao M.M., Linear functionals on Orlicz spaces: General theory, Pacific J. Math. 25 3 (1968), 553-585. (1968) Zbl0164.43601MR0412791
  33. Rao M.M., Ren Z.D., Theory of Orlicz Spaces, Marcel Dekker, New York, Basel, Hong Kong, 1991. Zbl0724.46032MR1113700
  34. Rolewicz S., Some remarks on the spaces N ( L ) and N ( l ) , Studia Math. 18 (1959), 1-9. (1959) Zbl0085.32301MR0103413
  35. Turpin P., Convexités dans les espaces vectoriels topologiques généraux, Dissertationes Math. 131 (1976). (1976) Zbl0331.46001MR0423044
  36. Wilansky A., Modern methods in topological vector spaces, McGrawhill, Inc., 1978. Zbl0395.46001MR0518316
  37. Zaanen A.C., Riesz Spaces II, North-Holland Publ. Com., Amsterdam, New York, Oxford, 1983. Zbl0519.46001MR0704021

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