Characterizations of Kadec-Klee properties in symmetric spaces of measurable functions
Vladimir ChilinDepartment of Mathematics, Tashkent State University, Tashkent 700095, UzbekistanPeter G. DoddsDepartment of Mathematics and Statistics, The Flinders University of South Australia, GPO Box 2100, Adelaide, SA 5001, AustraliaA. A. SedaevFedor SukochevDepartment of Mathematics and Statistics, The Flinders University of South Australia, GPO Box 2100, Adelaide, SA 5001, Australia
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Аннотация
We present several characterizations of Kadec-Klee properties in symmetric function spaces on the half-line, based on the <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper K"> <mml:semantics> <mml:mi>K</mml:mi> <mml:annotation encoding="application/x-tex">K</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -functional of J. Peetre. In addition to the usual Kadec-Klee property, we study those symmetric spaces for which sequential convergence in measure (respectively, local convergence in measure) on the unit sphere coincides with norm convergence.
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