We know that the tensor product of $L^1(G)$ and a Banach space $A$ is isometric to $L^1(G, A)$, the space of all Bochner-integrable $A$-valued functions on a locally compact group $G$. I am looking for a proof of this fact but I cannot find it. Does anyone have proof of it?
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$\begingroup$ Also asked here at MSE: math.stackexchange.com/questions/1397872/…. $\endgroup$– Cameron WilliamsCommented Aug 15, 2015 at 17:49
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$\begingroup$ Would you please confirm that my answer is fine? $\endgroup$– Tomasz KaniaCommented Sep 1, 2015 at 12:01
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$\begingroup$ @hosain: if you are satisfied with the answer then you should click the "tick" symbol to mark the question as "solved", so that it does not stay marked as "unsolved" $\endgroup$– Yemon ChoiCommented Sep 9, 2015 at 22:16
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1 Answer
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Of course you mean the projective tensor product. See Example 2.19 on p. 29. in
R. A. Ryan. Introduction to tensor products of Banach spaces. Springer Monographs in Mathematics. Springer-Verlag London Ltd., London, 2002.
answered Aug 15, 2015 at 12:31
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$\begingroup$ Excuse me Tomek thanks for your answer. That is fine. $\endgroup$– hosainCommented Sep 7, 2015 at 6:14