# Riesz depiction and also vector - valued features

A variation of the Riesz Representation Theorem claims that a continual straight useful on the room of continual actual - valued mappings on a portable statistics room, $C(X)$, can be understood an authorized Borel action on the set $X$. Exist any kind of comparable outcomes when we change $C(X)$ by the room of continual features of $X$ (portable statistics) right into $Y$ when (1) $Y=R^N$ or as a whole (2) $Y$ is a Banach room? I believe the solution is of course, yet I would love to locate the appropriate reference to start considering. Many thanks.

Yes, there are comparable cause the vector - valued instance. Dunford and also Schwartz is a typical reference for this example. for more details see this

Some symbols : $X$ is a set portable Hausdorff room. For a Banach room $Y$, the room of continual features from $X$ to $Y$, gifted with the supremum standard originating from the standard of $Y$, I represent by $C(X,Y)$. For a Banach room $Z$, I represent its twin by $Z'$.

Below is one means to think of the twin of $C(X,Y)$ for $Y$ a Banach room.

A component $\phi$ of $C(X,Y)'$ generates a family members of actions on $X$ parametrized by $Y$ in the list below means. Dealing with $\xi \in Y$, one can specify a straight useful $L_{\phi,\xi}$ on $C(X)$ by sending out the function $f$ on $X$, to the value of $\phi$ on the function $X \to Y$ offered by $x \mapsto f(x) \xi$. In icons : $$ L_{\phi,\xi}(f) = \phi(x \mapsto f(x) \xi). $$ From the common Riesz theory, there is after that an action $m_{\phi, \xi}$ specified on the Borel parts of $X$ pleasing $$ L_{\xi,\phi}(f) = \int_X f \, dm_{\phi, \xi}. $$ So from $\phi$ we have actually generated a family members of actions on $X$, one for each and every $\xi$ in $Y$.

Currently specify a map $m_{\phi}$ from the Borel parts of $X$ to $Y'$ as adheres to : for any kind of Borel part $E$ of $X$, specify $m_{\phi}(E)$ to be the straight useful on $Y$ offered by $$ m_{\phi}(E)(\xi) = \int_E 1 \, dm_{\phi, \xi}. $$ The map $m_{\phi}$ has numerous wonderful buildings (it is a $Y'$ - valued analogue of a normal authorized Borel action on $X$). Given that the features of the kind $x \mapsto f(x) \xi$, with $f \in C(X)$ and also $\xi \in Y$, are thick in $C(X,Y)$, it is very easy to show that $\phi$ is distinctly established by $m_{\phi}$. (The instinct is to consider $\phi$ as originating from $m_{\phi}$ as adheres to : for each and every $f \in C(X,Y)$, the number $\phi(f)$ is gotten "by incorporating, over $X$, the values of $f$ relative to the $Y'$ - valued action $m_{\phi}$, to make sure that $\phi(f) = \int_X f \, dm_{\phi}$. " You can consider this equally as an official point, or, assume sufficient concerning the assimilation of vector - valued features relative to vector - valued set mappings like $m_{\phi}$ to define this and also remove the quote marks.)

Anyhow, you can reverse this entire chain of thinking : beginning with a map from the Borel collections of $X$ to $Y'$ with wonderful adequate buildings, you can show that it has to be $m_{\phi}$ for some $\phi$ in $C(X,Y)'$. There is an all-natural idea of standard for these points (a "variation" standard) and also it ends up to accompany the standard you would certainly obtain from $C(X,Y)'$. So the twin of $C(X,Y)$, in this image, is a room of perfectly acted $Y'$ - valued mappings on the Borel parts of $X$, with a particular variant standard. When $Y$ is the scalars this is develops into the initial Riesz theory.

Extra usually you can consider any kind of bounded straight map from $C(X,Y)$ right into a Banach room $Z$ in comparable terms, yet points get extra difficult (the "measure - like" points you incorporate over $X$ to stand for maps $C(X,Y) \to Z$ take values in the straight drivers from $Y$ to $Z''$). You can additionally damage numerous theories below (as an example you can go down the density theory on $X$, or change $Y$ with an extra basic topological straight room, gave that you agree to make added difficult theories in order to mention a suitable theory).

There is an additional instructions you can go. If $X$ is portable Hausdorff and also $Y$ is a Banach room, the room $C(X,Y)$ is isometrically isomorphic to a particular tensor item, particularly the *injective Banach room tensor item *, of $C(X)$ with $Y$. So recognizing the twin of $C(X,Y)$ is a grandfather clause of recognizing the twin of an injective tensor item $A \otimes_i B$ of Banach rooms $A$ and also $B$. The twin of this tensor item has numerous characterizations. One remains in regards to Borel actions on the Cartesian item of the portable topological rooms $(A')_1$ and also $(B')_1$ (the device rounds of the duals of $A$ and also $B$, offered the weak - $*$ geography). Any kind of publication on Banach rooms that reviews the tensor item concept will certainly have theories concerning the injective Banach room tensor item and also just how duality connects with it.

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