Riesz representation and vector-valued functions

Question

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.

Answer 1

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

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.