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Mar 15, 2020 at 20:16 comment added Dirk Werner That the dual of $C(K,X)$ is the space of $X^*$-valued vector measures is a theorem due to Ivan Singer whose proof does not depend on the theory of tensor products; the original paper is in Russian, see Zbl 0087.31601
Mar 15, 2020 at 14:23 vote accept Yaddle
Mar 15, 2020 at 9:29 history answered Matthew Daws CC BY-SA 4.0