# Dual space of continuous Banach-space-valued functions

Let $$X$$ be a Banach space and $$K$$ some compact Hausdorff space. I am interested in the dual space of the Banach space $$C(K; X) = \lbrace f: K \to X, \ f \text{ is continuous}\rbrace, \qquad \lVert f \rVert_\infty := \max_{x \in K} \lVert f(x) \rVert.$$ In the case, $$X = \mathbb C$$, it is well known that the dual space is given by the space of all regular Borel measures $$\operatorname{rca}(K)$$ on the space $$K$$.

Now I think one can not hope, in the general case that $$X$$ is a Banach space at least, for this characterization to carry over to the vector-valued case as one needs some kind of lattice structure to define regularity for vector valued measures. Moreover, if one looks at the $$L^p$$-situation, $$1 \leq p <\infty$$, one has $$L^p(\Omega; X)' = L^{p'}(\Omega; X')$$ if and only if the space $$X$$ has the Radon-Nikodym property. So a natural conjecture for me would be something like the following:

If $$X$$ is a Banach lattice, maybe with some additional Banach space or lattice properties, e.g., Radon-Nikodym or order continuity of the norm, then one has $$C(K; X)' = \operatorname{rca}(K; X')$$, where $$\operatorname{rca}(K; X)$$ denotes the space of $$X'$$-valued regular Borel measures.

I would expect that people asked and solved this question already. So my question is if a result of this kind of flavour is known? Moreover, are there good references for this kind of results? Thanks in advance!

• If we consider the dual of $X$ as a complete locally convex space with the finest such topology that agrees with the weak $\ast$ topology on the unit ball, then the dual space you are looking for is just the space of bounded measures on $K$ with values in $X’$ which are Radon for this topology. – user131781 Mar 15 '20 at 11:34

The natural language to work in here is that of tensor norms. Here I follow Ryan, Introduction to tensor products of Banach spaces. Section 3.2 shows we can identify $$C(K;X)$$ with the injective Banach space tensor product $$C(K) \check\otimes X$$.

Thus, we are lead to understand the dual space of $$C(K) \check\otimes X$$. This (in a lot more generality) can be identified with the integral operators from $$C(K)$$ to $$X^*$$, see section 3.5.

Finally, we wish to understand integral operators. Chapter 5 does this. It turns out that every weakly compact operator $$T:C(K)\rightarrow X^*$$ can be understood as a vector measure $$\mu:K\rightarrow X^*$$ given by $$T(f) = \int_K f(x) \ d\mu(x)$$. Of course, we need to know what a vector measure is to fully understand this. Then Proposition 5.28 shows that $$T$$ is furthermore integral (all integral operators are weakly compact) exactly when $$\mu$$ has bounded variation.

So, with suitable definitions, the answer is essentially "yes" it all works. You ask about "regularity". For this, see Lemma 5.24. Hahn-Banach, and weak compactness, basically rescues you.

You might also look at the book by Diestel and Uhl, "Vector Measures".

• 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 – Dirk Werner Mar 15 '20 at 20:16