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A version of the Riesz Representation Theorem says that a continuous linear functional on the space of continuous real-valued mappings on a compact metric space, $C(X)$, can be identified with a signed Borel measure on the set $X$. Are there any similar results when we replace $C(X)$ by the space of continuous functions of $X$ (compact metric) into $Y$ when (1) $Y=R^N$ or in general (2) $Y$ is a Banach space? I suspect the answer is yes, but I would like to find the right reference to start looking at. Thanks.

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In the case of $\mathbb R^N$, the answer is certainly "yes", since you can just apply the usual Riesz rep. thm. component by component. –  Matt E Dec 8 '10 at 6:18

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Yes, there are similar results in the vector-valued case. Dunford and Schwartz is a standard reference for this kind of thing.

Some notation: $X$ is a fixed compact Hausdorff space. For a Banach space $Y$, the space of continuous functions from $X$ to $Y$, endowed with the supremum norm coming from the norm of $Y$, I denote by $C(X,Y)$. For a Banach space $Z$, I denote its dual by $Z'$.

Here is one way to think about the dual of $C(X,Y)$ for $Y$ a Banach space.

An element $\phi$ of $C(X,Y)'$ gives rise to a family of measures on $X$ parametrized by $Y$ in the following way. Fixing $\xi \in Y$, one can define a linear functional $L_{\phi,\xi}$ on $C(X)$ by sending the function $f$ on $X$, to the value of $\phi$ on the function $X \to Y$ given by $x \mapsto f(x) \xi$. In symbols: $$ L_{\phi,\xi}(f) = \phi(x \mapsto f(x) \xi). $$ From the usual Riesz theorem, there is then a measure $m_{\phi, \xi}$ defined on the Borel subsets of $X$ satisfying $$ L_{\xi,\phi}(f) = \int_X f \, dm_{\phi, \xi}. $$ So from $\phi$ we have produced a family of measures on $X$, one for each $\xi$ in $Y$.

Now define a map $m_{\phi}$ from the Borel subsets of $X$ to $Y'$ as follows: for any Borel subset $E$ of $X$, define $m_{\phi}(E)$ to be the linear functional on $Y$ given by $$ m_{\phi}(E)(\xi) = \int_E 1 \, dm_{\phi, \xi}. $$ The map $m_{\phi}$ has various nice properties (it is a $Y'$-valued analogue of a regular signed Borel measure on $X$). Since the functions of the form $x \mapsto f(x) \xi$, with $f \in C(X)$ and $\xi \in Y$, are dense in $C(X,Y)$, it is easy to show that $\phi$ is uniquely determined by $m_{\phi}$. (The intuition is to think of $\phi$ as coming from $m_{\phi}$ as follows: for each $f \in C(X,Y)$, the number $\phi(f)$ is obtained "by integrating, over $X$, the values of $f$ with respect to the $Y'$-valued measure $m_{\phi}$, so that $\phi(f) = \int_X f \, dm_{\phi}$." You can think of this just as a formal thing, or, think enough about the integration of vector-valued functions with respect to vector-valued set mappings like $m_{\phi}$ to formalize this and remove the quotation marks.)

Anyway, you can reverse this whole chain of reasoning: starting with a map from the Borel sets of $X$ to $Y'$ with nice enough properties, you can show that it must be $m_{\phi}$ for some $\phi$ in $C(X,Y)'$. There is a natural notion of norm for these things (a "variation" norm) and it turns out to coincide with the norm you'd get from $C(X,Y)'$. So the dual of $C(X,Y)$, in this picture, is a space of nicely behaved $Y'$-valued mappings on the Borel subsets of $X$, with a certain variation norm. When $Y$ is the scalars this is turns into the original Riesz theorem.

More generally you can think of any bounded linear map from $C(X,Y)$ into a Banach space $Z$ in similar terms, but things get more complicated (the "measure-like" things you integrate over $X$ to represent maps $C(X,Y) \to Z$ take values in the linear operators from $Y$ to $Z''$). You can also weaken various hypotheses here (e.g. you can drop the compactness hypothesis on $X$, or replace $Y$ with a more general topological linear space, provided that you are willing to make additional complicated hypotheses in order to state a decent theorem).

There is another direction you can go. If $X$ is compact Hausdorff and $Y$ is a Banach space, the space $C(X,Y)$ is isometrically isomorphic to a certain tensor product, namely the injective Banach space tensor product, of $C(X)$ with $Y$. So identifying the dual of $C(X,Y)$ is a special case of identifying the dual of an injective tensor product $A \otimes_i B$ of Banach spaces $A$ and $B$. The dual of this tensor product has various characterizations. One is in terms of Borel measures on the Cartesian product of the compact topological spaces $(A')_1$ and $(B')_1$ (the unit balls of the duals of $A$ and $B$, given the weak-$*$ topology). Any book on Banach spaces that discusses the tensor product theory will have theorems about the injective Banach space tensor product and how duality interacts with it.

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A very minor suggestion: as this website is target at a more general audience of individuals interested in mathematics, and not necessarily those who make a career out of it, it may be helpful to include also the title of Dunford and Schwartz's classic book, Linear Operators, in the answer. After all, the casual reader (or possibly even the original poster) may not be familiar with it. –  Willie Wong Dec 8 '10 at 9:36
    
I found it very interesting. I would greatly appreciate if you or anyone else could give me a more precise reference (I am assuming you meant Linear Operators Part I, right?). Thanks again in any case. –  Kyle Wilson Dec 8 '10 at 19:59
    
I looked; maybe it's not quite in Dunford and Schwartz. (In VI.7 of Vol. 1, Theorems 2 and 3, operators from $C(X) \to Z$ are represented as above; the details are also in their paper "Weak compactness and vector measures" in the Canadian Journal of Math, fully available on Google books.) But the generalization to $C(X,Y)$ is not there; maybe it is not due to them. It seems to be due to Foias and Singer (a paper "Some remarks on the representation of linear operators..." in 1960). But the paper "Linear operators and vector measures" by Brooks and Lewis contains a more general theorem... –  anon Dec 9 '10 at 2:26
    
and is publically available at ams.org/journals/tran/1974-192-00/S0002-9947-1974-0338821-5/… (Theorem 2.2 is even more general than you need). It is worth pointing out that all of the technicalities arising here are largely when the space $Y$ is infinite dimensional. As Matt E pointed out, if $Y$ is finite dimensional, the dual must be in some sense just a "number of copies" of the dual of $C(X)$, at least as a vector space, and the only difficulty is to identify the right norm for it. –  anon Dec 9 '10 at 2:31

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