A characterization of the Mackey topology $\tau (L^ \infty , L^ 1)$
HTML articles powered by AMS MathViewer
- by Marian Nowak PDF
- Proc. Amer. Math. Soc. 108 (1990), 683-689 Request permission
Abstract:
We give a description of the Mackey topology $\tau ({L^\infty },{L^1})$ for finite measures in terms of a family of norms defined by certain Young functions. As an application we obtain various topological characterizations of sequential convergence in $\tau ({L^\infty },{L^1})$. Moreover, we obtain a criterion for relative weak compactness in ${L^1}$ in terms of the integral functional defined by some Young function.References
- Charalambos D. Aliprantis and Owen Burkinshaw, Locally solid Riesz spaces, Pure and Applied Mathematics, Vol. 76, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1978. MR 0493242
- Tsuyoshi Andô, Weakly compact sets in Orlicz spaces, Canadian J. Math. 14 (1962), 170–176. MR 157228, DOI 10.4153/CJM-1962-012-7
- James Bell Cooper, Saks spaces and applications to functional analysis, Notas de Matemática [Mathematical Notes], vol. 64, North-Holland Publishing Co., Amsterdam-New York, 1978. MR 492535
- Nelson Dunford and Jacob T. Schwartz, Linear Operators. I. General Theory, Pure and Applied Mathematics, Vol. 7, Interscience Publishers, Inc., New York; Interscience Publishers Ltd., London, 1958. With the assistance of W. G. Bade and R. G. Bartle. MR 0117523
- L. V. Kantorovich and G. P. Akilov, Funktsional′nyĭ analiz, 3rd ed., “Nauka”, Moscow, 1984 (Russian). MR 788496
- M. A. Krasnosel′skiĭ and Ja. B. Rutickiĭ, Convex functions and Orlicz spaces, P. Noordhoff Ltd., Groningen, 1961. Translated from the first Russian edition by Leo F. Boron. MR 0126722 W. A. Luxemburg, Banach function spaces, Delft, 1955.
- W. A. J. Luxemburg and A. C. Zaanen, Compactness of integral operators in Banach function spaces, Math. Ann. 149 (1962/63), 150–180. MR 145374, DOI 10.1007/BF01349240
- Marian Nowak, On the finest Lebesgue topology on the space of essentially bounded measurable functions, Pacific J. Math. 140 (1989), no. 1, 155–161. MR 1019072, DOI 10.2140/pjm.1989.140.155
- K. D. Stroyan, A characterization of the Mackey uniformity $m(L^{\infty },$ $L^{1})$ for finite measures, Pacific J. Math. 49 (1973), 223–228. MR 336315, DOI 10.2140/pjm.1973.49.223
- A. Wiweger, Linear spaces with mixed topology, Studia Math. 20 (1961), 47–68. MR 133664, DOI 10.4064/sm-20-1-47-68
Additional Information
- © Copyright 1990 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 108 (1990), 683-689
- MSC: Primary 46E30; Secondary 46A20
- DOI: https://doi.org/10.1090/S0002-9939-1990-0991705-6
- MathSciNet review: 991705