# Riesz representation theorem for vector-valued fields

Let $Q$ be a locally compact Hausdorff space, and let $V$ be a topological vector space. Consider the space $X = C_0(Q, V)$ of $V$-valued fields which vanish at infinity. Let $X^*$ denote the dual space of continuous linear functionals of $X$, and let $M = M(Q,V)$ denote the space of regular $V$-valued Borel measures on $Q$.

Under what conditions on the space $V$ is the space of measures $M$ isomorphic to the dual space $X^*$?

Is there a good reference for general Riesz representation theorems? Preferably one which takes a category-theoretic approach.

-
Umm... do you want $M$ to be $V$ valued or $V^*$ valued? – Willie Wong Mar 18 '13 at 13:03
The best (relatively recent) reference for this circle of ideas is the classic "Vector measures" by Diestel and Uhl and I would recommend this as a starting point. The standard treatise of Dunford and Schwartz also contains relevant material. If $V$ is a Banach space and $Q$ is compact, one can indeed get a result of this form using methods of category theory by using the fact that every Banach space is an inductive limit of its finite dimensional subspaces and then dualising---for a suitable definition of measures with values in a dual Banach space. This can then be generalised. – jbc Mar 18 '13 at 14:33
@jbc: What do you mean by every Banach space is an inductive limit of its finite dimensional subspaces? In which category? Shouldn't the inductive limit $X=\lim X\alpha$ have the universal property that a linear map $X\to Y$ is continuous (a morphism of the category) iff all restrictions to $X_\alpha$ are continuous? If all $X_\alpha$ are finite dimensional this is no condition. – Jochen Wengenroth Mar 19 '13 at 7:27
I have two remarks: 1) Do you already know the answer if $V$ is an abstract finite-dimensional vector space? 2) Regarding Willie's question: If you integrate a $V$-valued function times a $V$-valued measure over $Q$, you don't get a real number, which is necessary in order for $M$ to be naturally isomorphic to $X^*$. At best you get an element in $V\otimes V$. – Deane Yang Mar 23 '13 at 14:58
– Deane Yang Mar 23 '13 at 15:00

Since my comment seems to have been misunderstood, I would like to take the opportunity to expand on it. The basic categories are $\bf {Ban}_1$ and $\bf W$ of Waelbroeck spaces, i.e., Banach spaces with linear contractions as morphisms, resp., Banach spaces provided with an additional compact, linear topology on the unit ball (details can be found in the book by Cigler, Losert and Michor on categories of Banach spaces). If $E$ is a Banach space, then its dual $E'$ is a Waelbroeck space and indeed the two categories are dual to each other. The important example for us will be the pair $C(K)$ of continuous functions on a compactum and its dual, the space $M(K)$ of Radon measures thereon. The latter has, as does every Waelbroeck space, a natural complete, locally convex topology---the finest to agree with the given compact one on the unit ball---and in our example, this (and not the norm) is usually the natural one. We now denote the family of finite dimensional subspaces of a Banach space $E$ by $\cal F$ and use the fact that $E$ is the inductive limit of this family, regarded as an inductive spectrum in $\bf {Ban_1}$ in the natural way (this is very simple and can be found explicitly in the above reference). It follows fairly easily that $C(K;E)$, the Banach space of continuous functions with values in $E$, can be identified with the inductive limit of the spectrum $\{C(K;F), F \in \cal F\}$. General abstract nonsense shows that the dual of the latter is the projective limit (in the sense of the category $\bf W$) of $\{M(K;F') : F \in \cal F\}$ (we are using the trivial extension of the Riesz representation theorem to the case of functions with values in a finite dimensional space). One can then identify the elements of this projective limit with measures which take their values in $E'$ and which are bounded and Radon for the topology mentioned above to obtain the desired representation of the dual of $C(K;E)$.
If one is prepared to use the extension of the Riesz representation theorem which covers the case of a completely regular space $S$ and identifies the space of bounded, Radon measures on $S$ as the dual of the space $C^b(S)$ of bounded continuous functions with the strict topology (see the monograph "Saks spaces and Applications to Functional Analysis"), then one can obtain a suitable version of this duality which works for completely regular spaces.