# Banach

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Name        Banach
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| Feb4 | comment  | Riesz representation for an infinite-dimensional spaceI looked up projective limit, but I don't see why $C(X)$ is the projective limit of $C(K)$. Would you please indicate what are the coordinate spaces and the bonding maps in the inverse limit?                                                                                                                                                                                                                                                                                                        |
| Feb4 | awarded  | ● Commentator                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                |
| Feb4 | comment  | Riesz representation for an infinite-dimensional spaceAccording to my topology book, there is a restriction. The statement, "a function on X is continuous iff its restrictions on compact subsets are continuous" requires X to be compactly generated. However, in my case X is compactly generated.                                                                                                                                                                                                                                                       |
| Feb4 | comment  | Riesz representation for an infinite-dimensional space@Yemon, which smaller space do you suggest to consider instead of $C(X)$? Do you know of a reference for Riesz representation for $C(X)$ where $X$ is a metrizable non LCH space?                                                                                                                                                                                                                                                                                                                      |
| Feb4 | comment  | Riesz representation for an infinite-dimensional space What does it mean that $C(X)$ is the projective limit of $C(K)$? Does this require X to be a countable union of compact sets?                                                                                                                                                                                                                                                                                                                                                                         |
| Feb4 | comment  | Riesz representation for an infinite-dimensional space@Yemon, You are misunderstanding the question. I give you a Banach algebra $X$ that is infinite dimensional. I am asking for a topology on $C(X)$ and a characterization of its dual. I am asking if anyone has seen such sort of theorem. The focus is the infinite-dimensionality of $X$. Either you have seen a Riesz-type theorem for infinite-dimensional spaces or not. If you don't like $X$ being an algebra, assume it's just a Banach space, or assume it's just a topological vector space. |
| Feb3 | comment  | Riesz representation for an infinite-dimensional space$C(X)$ can be given a topology (most likely not a normable), but I don't want to specify one because that is part of the question. I am asking if there is some sort of Riesz representation theorem for an infinite-dimensional space $X$ where $C(X)$ has some nontrivial topology?                                                                                                                                                                                                                  |
| Feb3 | comment  | Riesz representation for an infinite-dimensional spaceIs $C(X)$ suppose to be a Banach space?                                                                                                                                                                                                                                                                                                                                                                                                                                                                |
| Feb3 | comment  | Riesz representation for an infinite-dimensional spaceI didn't say $C(X)$ is a Banach space.                                                                                                                                                                                                                                                                                                                                                                                                                                                                 |
| Feb3 | asked    | Riesz representation for an infinite-dimensional space                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                       |
| Dec4 | comment  | Absolutely Continuous Invariant Measures for Piecewise Convex MapsYou can get away with $C^{1+\alpha}$, but with less regularity, I'm not sure.                                                                                                                                                                                                                                                                                                                                                                                                              |
| Dec3 | awarded  | ● Editor                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     |
| Dec3 | revised  | Absolutely Continuous Invariant Measures for Piecewise Convex Mapsadded 341 characters in body                                                                                                                                                                                                                                                                                                                                                                                                                                                               |
| Dec3 | answered | Absolutely Continuous Invariant Measures for Piecewise Convex Maps                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           |
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