Timeline for Extending a Hilbert space isometrically
Current License: CC BY-SA 3.0
9 events
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| Mar 16, 2013 at 10:52 | comment | added | Jochen Wengenroth | @Tom: Even if there is any metric on $F$ inducing the vector space topology such that $f$ is an isometry then $f(H)$ will be already a completely metrizable topological vector space. This not obvious (since the uniformities may be a priori different) but true because of a theorem of Victor Klee (solving a problem of Banach). Therefore, $F=f(H)$ still holds. | |
| Mar 15, 2013 at 22:02 | comment | added | Marcel Bischoff | Aren't Gelfand triples giving examples of what you want ncatlab.org/nlab/show/Gelfand+triple | |
| Mar 15, 2013 at 19:52 | comment | added | paul garrett | Are the inclusions of Levi-Sobolev spaces $H^s(T^n)\rightarrow H^t(T^n)$ on products $T^n$ of circles, for $s>t$, the kind of example you want? If so, it illustrates that a Hilbert-space structure can get weaker... | |
| Mar 15, 2013 at 18:49 | comment | added | Tom LaGatta | @Jochen Wengenroth: I specifically made no additional assumptions on the Hilbert space nor the larger topological space $X$, such as separability or local convexity. Your remark on the separable case is interesting, and I thank you for making the point. For the second question, I mean to say, "when does F admit the structure of a Fréchet space?" Certainly, if its topology is completely metrizable, then the metric on $F$ will be an extension of the metric from $H$. | |
| Mar 15, 2013 at 18:46 | history | edited | Tom LaGatta | CC BY-SA 3.0 |
added 25 characters in body
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| Mar 15, 2013 at 16:18 | comment | added | jbc | Since you can always embed $H$ isometrically into $H \times Y$ for any topological vector space $Y$, there would seem to be plenty of scope. | |
| Mar 15, 2013 at 14:46 | comment | added | Jochen Wengenroth | ...normed and $f$ is an isometry then $f(H)$ is complete and hence closed in $X$ (if you assume that $X$ is Hausdorff). | |
| Mar 15, 2013 at 14:45 | comment | added | Jochen Wengenroth | The obvious idea to embed the separable Hilbert space into a sequentially complete locally convex space is to find a sequence $x_n$ which converges fast to $0$ and to define $f:\ell_2 \to X$ by $f((a_n)_{n\in\mathbb N})= \sum\limits_{n=1}^\infty a_n x_n$. This map will be injective if the sequence is topologically linearly m-independent$. This is discussed on page 37 in the book *Barrelled locally convex spaces of Bonet and Perez-Carreras. The second question is not clear to me: If there is no norm on the Frechet space $F$, what do you mean by isometric? On the other hand, if $F$ is ... | |
| Mar 15, 2013 at 12:53 | history | asked | Tom LaGatta | CC BY-SA 3.0 |