Summary
We generalize the classical Riesz convergence theorem to Lorentz spaces, i.e., if <Emphasis Type=”Italic”>f,f</Emphasis><Subscript>1</Subscript><Emphasis Type=”Italic”>, f</Emphasis><Subscript>2</Subscript><Emphasis Type=”Italic”>,..</Emphasis>e<Emphasis Type=”Italic”>.</Emphasis> <Emphasis Type=”Italic”>L<Superscript>p,q</Superscript></Emphasis> such that <Emphasis Type=”Italic”>f<Subscript>n</Subscript> </Emphasis>→<Emphasis Type=”Italic”>f</Emphasis> (a.e. or in measure) and ׀׀<Emphasis Type=”Italic”>f<Subscript>n</Subscript></Emphasis>׀׀<Emphasis Type=”Italic”><Subscript>p,q</Subscript></Emphasis> → ׀׀<Emphasis Type=”Bold”>f</Emphasis> ׀׀<Emphasis Type=”Italic”><Subscript>p,q</Subscript></Emphasis>, then ׀׀<Emphasis Type=”Italic”>f<Subscript>n</Subscript></Emphasis> - <Emphasis Type=”Italic”>f</Emphasis>׀׀<Emphasis Type=”Italic”><Subscript>p,q </Subscript></Emphasis>→ 0 as <Emphasis Type=”Italic”>n</Emphasis> →∞.
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Kohli, J., Singh, D. Between compactness and quasicompactness. Acta Math Hung 106, 317–329 (2005). https://doi.org/10.1007/s10474-005-0022-4
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DOI: https://doi.org/10.1007/s10474-005-0022-4