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made one statement more precise
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jbc
jbc
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May I add some information on this topic? Firstly, the space $C(X)$ is not usually a Frechet space---you need some countability condition on the compact subsets of $X$, e.g., it being $\sigma$-compact and locally compact. It is not even complete in the general case---for that you need the condition that it be a $k_R$-space. The dual of $C(X)$ can be identified, with the aid of some abstract locally convex theory and the RRT for compact spaces, with the space of measures on $K$ with compact support (i.e. those arising from measures on some compact subset in the natural way). If $X$ is locally compact, then Bourbaki used the dual of the space of continuous functions with compact support as the {\it definition} of the space of (unbounded) measures on $X$. One can then interpret its members as measures in the classical sense (i.e. as functions defined on a suitable class of sets) by the usual extension methods. I would suggest that the most useful extension of the Riesz representation theorem is the one for bounded, Radon measures on a (completely regular) space. For this one has to go beyond the more common classes of Banach or even locally convex spaces, something which was done by Buck in the 50's. He introduced a locally convex topology on $C^b(X)$ (the bounded, continuous functions) using weighted seminorms for which exactly the kind of representation theorem one would expect and hope for obtains. He did this for locally compact spaces but it was soon extended to the general case, using the methods of mixed topologies and Saks spaces of the polish school. There are many indications that this is the correct structure---the natural versions of the Stone-Weierstrass theorem hold for it and its spectrum (regarding $C^b(X)$ as an algebra) is identifiable with $X$ so that one has a form of the Gelfand-Naimark theory. Further indications of its suitability are that if one considers generalised spectra, i.e., continuous, algebraic homomorphisms into more general algebras then one obtains interesting results and concepts. The important case is where $A$ is $L(H)$ (or, more generally, a von Neumann algebra). One then gets spaces of observables (in the sense of quantum theory) in the case where the underlying topological space is the real line and this provides them in a natural way with a structure which opens a path to a natural and rigorous approach to analysis in the context of spaces of observables---distributions, analytic functions, ...).

May I add some information on this topic? Firstly, the space $C(X)$ is not usually a Frechet space---you need some countability condition on the compact subsets of $X$, e.g., it being $\sigma$-compact and locally compact. It is not even complete in the general case---for that you need the condition that it be a $k_R$-space. The dual of $C(X)$ can be identified, with the aid of some abstract locally convex theory and the RRT for compact spaces, with the space of measures on $K$ with compact support (i.e. those arising from measures on some compact subset in the natural way). If $X$ is locally compact, then Bourbaki used the dual of the space of continuous functions with compact support as the {\it definition} of the space of (unbounded) measures on $X$. One can then interpret its members as measures in the classical sense (i.e. as functions defined on a suitable class of sets) by the usual extension methods. I would suggest that the most useful extension of the Riesz representation theorem is the one for bounded, Radon measures on a (completely regular) space. For this one has to go beyond the more common classes of Banach or even locally convex spaces, something which was done by Buck in the 50's. He introduced a locally convex topology on $C^b(X)$ (the bounded, continuous functions) using weighted seminorms for which exactly the kind of representation theorem one would expect and hope for obtains. He did this for locally compact spaces but it was soon extended to the general case, using the methods of mixed topologies and Saks spaces of the polish school. There are many indications that this is the correct structure---the natural versions of the Stone-Weierstrass theorem hold for it and its spectrum (regarding $C^b(X)$ as an algebra) is identifiable with $X$ so that one has a form of the Gelfand-Naimark theory. Further indications of its suitability are that if one considers generalised spectra, i.e., continuous, algebraic homomorphisms into more general algebras then one obtains interesting results and concepts. The important case is where $A$ is $L(H)$ (or, more generally, a von Neumann algebra). One then gets spaces of observables (in the sense of quantum theory) and this provides them in a natural way with a structure which opens a path to a natural and rigorous approach to analysis in the context of spaces of observables---distributions, analytic functions, ...).

May I add some information on this topic? Firstly, the space $C(X)$ is not usually a Frechet space---you need some countability condition on the compact subsets of $X$, e.g., it being $\sigma$-compact and locally compact. It is not even complete in the general case---for that you need the condition that it be a $k_R$-space. The dual of $C(X)$ can be identified, with the aid of some abstract locally convex theory and the RRT for compact spaces, with the space of measures on $K$ with compact support (i.e. those arising from measures on some compact subset in the natural way). If $X$ is locally compact, then Bourbaki used the dual of the space of continuous functions with compact support as the {\it definition} of the space of (unbounded) measures on $X$. One can then interpret its members as measures in the classical sense (i.e. as functions defined on a suitable class of sets) by the usual extension methods. I would suggest that the most useful extension of the Riesz representation theorem is the one for bounded, Radon measures on a (completely regular) space. For this one has to go beyond the more common classes of Banach or even locally convex spaces, something which was done by Buck in the 50's. He introduced a locally convex topology on $C^b(X)$ (the bounded, continuous functions) using weighted seminorms for which exactly the kind of representation theorem one would expect and hope for obtains. He did this for locally compact spaces but it was soon extended to the general case, using the methods of mixed topologies and Saks spaces of the polish school. There are many indications that this is the correct structure---the natural versions of the Stone-Weierstrass theorem hold for it and its spectrum (regarding $C^b(X)$ as an algebra) is identifiable with $X$ so that one has a form of the Gelfand-Naimark theory. Further indications of its suitability are that if one considers generalised spectra, i.e., continuous, algebraic homomorphisms into more general algebras then one obtains interesting results and concepts. The important case is where $A$ is $L(H)$ (or, more generally, a von Neumann algebra). One then gets spaces of observables (in the sense of quantum theory) in the case where the underlying topological space is the real line and this provides them in a natural way with a structure which opens a path to a natural and rigorous approach to analysis in the context of spaces of observables---distributions, analytic functions, ...).

Source Link
jbc
jbc
  • 2.4k
  • 1
  • 26
  • 11

May I add some information on this topic? Firstly, the space $C(X)$ is not usually a Frechet space---you need some countability condition on the compact subsets of $X$, e.g., it being $\sigma$-compact and locally compact. It is not even complete in the general case---for that you need the condition that it be a $k_R$-space. The dual of $C(X)$ can be identified, with the aid of some abstract locally convex theory and the RRT for compact spaces, with the space of measures on $K$ with compact support (i.e. those arising from measures on some compact subset in the natural way). If $X$ is locally compact, then Bourbaki used the dual of the space of continuous functions with compact support as the {\it definition} of the space of (unbounded) measures on $X$. One can then interpret its members as measures in the classical sense (i.e. as functions defined on a suitable class of sets) by the usual extension methods. I would suggest that the most useful extension of the Riesz representation theorem is the one for bounded, Radon measures on a (completely regular) space. For this one has to go beyond the more common classes of Banach or even locally convex spaces, something which was done by Buck in the 50's. He introduced a locally convex topology on $C^b(X)$ (the bounded, continuous functions) using weighted seminorms for which exactly the kind of representation theorem one would expect and hope for obtains. He did this for locally compact spaces but it was soon extended to the general case, using the methods of mixed topologies and Saks spaces of the polish school. There are many indications that this is the correct structure---the natural versions of the Stone-Weierstrass theorem hold for it and its spectrum (regarding $C^b(X)$ as an algebra) is identifiable with $X$ so that one has a form of the Gelfand-Naimark theory. Further indications of its suitability are that if one considers generalised spectra, i.e., continuous, algebraic homomorphisms into more general algebras then one obtains interesting results and concepts. The important case is where $A$ is $L(H)$ (or, more generally, a von Neumann algebra). One then gets spaces of observables (in the sense of quantum theory) and this provides them in a natural way with a structure which opens a path to a natural and rigorous approach to analysis in the context of spaces of observables---distributions, analytic functions, ...).