Timeline for What is the tensor product of $L^p(\bf R)$ with $L^q(\bf R)$?
Current License: CC BY-SA 2.5
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| Feb 6, 2010 at 21:23 | history | edited | Mark Meckes |
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| Feb 6, 2010 at 21:23 | answer | added | Mark Meckes | timeline score: 5 | |
| Feb 6, 2010 at 0:48 | comment | added | Yemon Choi | @Anonymous: It was your use of the definite article and the word "should" which I was quibbling with - depending on the problem at hand, one may have to use different norms, and so as it stands your question is hard to answer because of the lack of context. FWIW, it sounds like you want something lke one of the Chevet-Saphar tensor norms. R. Ryan has a more accessible though les far-reaching treatment of tensor norms, which might also be in your library or similar. | |
| Feb 6, 2010 at 0:41 | comment | added | Anonymous | @Martin: Although I'm obviously not an expert on these thinks, I'm well aware of the variety of possible norms on the tensor product. Part of my question is: Which one is the good definition in this setting? There should be a good notion in this case. In the $L^2$-case, we certainly want the Hilbert space notion of the tensor product, since this one gives a nice answer, and I expect that there is some setting which handles the $L^p \otimes L^q$-case quite well. @Yemon: Thanks. I'll take a look at this book. | |
| Feb 6, 2010 at 0:40 | comment | added | Johannes Hahn | If the tensorproduct of $L^2(\mathbb{R})$ with itself is $L^2(\mathbb{R}^2)$, then perhaps the tensorproduct you're looking for is something like "All f with $(\int (\|f(x,\cdot)\|_p)^q\,dx)^{1/q}=(\int (\|f(\cdot,y)\|_q)^p\,dy)^{1/p}<\infty$" | |
| Feb 6, 2010 at 0:32 | comment | added | Yemon Choi | For a hint at just how non-obvious any of the answers are (even for p=q=2$) I recommend a quick look at (the size of) Defant and Floret's bok "Tensor norms and operator ideals" | |
| Feb 6, 2010 at 0:25 | comment | added | Martin Brandenburg | there are several common norms on the tensor product of normed algebras. please specify. | |
| Feb 6, 2010 at 0:05 | history | asked | Anonymous | CC BY-SA 2.5 |