Topological equivalence in the space of integrable vector-valued functions

Gutman, Semion

doi:10.1090/S0002-9939-1985-0766523-2

Topological equivalence in the space of integrable vector-valued functions
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by Semion Gutman PDF

Proc. Amer. Math. Soc. 93 (1985), 40-42 Request permission

Abstract:

The Banach space ${L^1}(0,T;X)$ is retopologized by $|||f||| = \max ||\int _a^bfdt||$, $0 \leqslant a \leqslant b \leqslant T$, where $||.||$ is the norm in the given Banach space $X$. It is shown here that this topology coincides with the usual weak topology of ${L^1}(0,T;X)$ on a wide class of weakly compact subsets.

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