Weak sequential convergence in 𝐿_{𝐸}^{∞} and Dunford-Pettis property of 𝐿_{𝐸}¹

Khurana, Surjit Singh

doi:10.1090/S0002-9939-1980-0548089-0

Weak sequential convergence in $L_{E}^{\infty }$ and Dunford-Pettis property of $L_{E}^{1}$
HTML articles powered by AMS MathViewer

by Surjit Singh Khurana PDF

Proc. Amer. Math. Soc. 78 (1980), 85-88 Request permission

Abstract:

For a $\sigma$-finite measure space $(X,\mathfrak {A},\mu )$ it is proved that weak sequential convergence in $L_E^\infty$ implies almost everywhere pointwise convergence, with the weak topology on the Banach space E. Also it is proved that if weak and norm sequential convergence coincide in $E’$, then $L_E^1$ has the Dunford-Pettis property.

References

Jürgen Batt, On weak compactness in spaces of vector-valued measures and Bochner integrable functions in connection with the Radon-Nikodým property of Banach spaces, Rev. Roumaine Math. Pures Appl. 19 (1974), 285–304. MR 341081
Owen Burkinshaw, Weak compactness in the order dual of a vector lattice, Trans. Amer. Math. Soc. 187 (1974), 183–201. MR 394098, DOI 10.1090/S0002-9947-1974-0394098-6
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
H. Fakhoury, Sur les espaces de Banach ne contenant pas $l^{1}(\textbf {N})$, Math. Scand. 41 (1977), no. 2, 277–289 (French). MR 500085, DOI 10.7146/math.scand.a-11720
A. Grothendieck, Sur les applications linéaires faiblement compactes d’espaces du type $C(K)$, Canad. J. Math. 5 (1953), 129–173 (French). MR 58866, DOI 10.4153/cjm-1953-017-4
A. Grothendieck, Sur certains sous-espaces vectoriels de $L^p$, Canad. J. Math. 6 (1954), 158–160 (French). MR 58867, DOI 10.4153/cjm-1954-017-x
A. Ionescu Tulcea and C. Ionescu Tulcea, Topics in the theory of lifting, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 48, Springer-Verlag New York, Inc., New York, 1969. MR 0276438, DOI 10.1007/978-3-642-88507-5
A. Ionescu Tulcea, Liftings compatible with topologies, Bull. Soc. Math. Grèce (N.S.) 8 (1967), no. fasc. 1, 116–126. MR 230875
A. L. Peressini, Banach limits in vector lattices, Studia Math. 35 (1970), 111–121. MR 265912, DOI 10.4064/sm-35-2-111-121
Prakash Pethe and Nimbakrishna Thakare, Note on Dunford-Pettis property and Schur property, Indiana Univ. Math. J. 27 (1978), no. 1, 91–92. MR 458123, DOI 10.1512/iumj.1978.27.27008