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Notes on interpolation by the real method between $ C(T,A\sb 0)$ and $ C(T,A\sb 1)$


Author: Mieczysław Mastyło
Journal: Proc. Amer. Math. Soc. 102 (1988), 945-948
MSC: Primary 46E40; Secondary 46M35
DOI: https://doi.org/10.1090/S0002-9939-1988-0934872-3
MathSciNet review: 934872
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Abstract: Let $ A$ be a Banach space and let $ T$ be a compact Hausdorff space. We denote by $ C(T,A)$ the Banach space of all $ A$-valued continuous functions defined on $ T$ endowed with the supremum norm. We show that if $ T$ is infinite and $ ({A_0},{A_1})$ is a Banach couple with $ {A_0}$ continuously embedded in $ {A_1}$, then the interpolation space $ {(C(T,{A_0}),C(T,{A_1}))_{\varphi ,p}}$ is equal to $ C\left( {T,{{\left( {{A_0},{A_1}} \right)}_{\varphi ,p}}} \right)$ if and only if $ {A_0}$ is closed in $ {A_1}$.


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DOI: https://doi.org/10.1090/S0002-9939-1988-0934872-3
Keywords: Interpolation spaces, complemented subspaces, $ K$-functional
Article copyright: © Copyright 1988 American Mathematical Society

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