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It is a known fact that any two classical Hilbert spaces are isomorphic, in particular, any classical Hilbert space is isomorphic to $L^2$.

Consider now weighted Hilbert spaces, that is, Hilbert spaces with a weighted inner product

$ \langle f, g \rangle = \int f \ g \ w \ dx $

where $w(x)$ is the weight function. This situation appears frequently in Sturm-Liouville eigenvalue problems. Let's call $H_w$ to such weighted Hilbert space.

My question is: are all weighted Hilbert spaces also isomorphic?. In other words, is any weighted Hilbert space $H_w$ isomorphic to $L^2_w$?

Thank you very much.

fjgg1549
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  • Consider $w(t) = 0$. The answer depends on the class where $w(t)$ lies. – Pavel Ievlev Dec 03 '17 at 09:51
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    So, the right question here would be "what are the conditions on $w(t)$ for $L_{2, , w}$ and $L_{2}$ to be isomorphic?". This question could be generalized: "what are the conditions on measures $\mu$ and $\nu$ for $L_{2, , \mu}$ and $L_{2, , \nu}$ to be isomorphic?" – Pavel Ievlev Dec 03 '17 at 09:57
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    It is not exactly that all Hilbert spaces are isomorphic. It depends on the "dimension" –  Dec 03 '17 at 09:58
  • Hilbert spaces can have a variety of inner products and also not be composed of functions. How, fundamentally, is a weighted Hilbert space differ from the a Hilbert space as defined by the axioms? – Theo Bendit Dec 03 '17 at 10:09
  • Sorry for not being specific. I am currently working on a Sturm-Liouville problem and my question refers to infinite dimensional functional spaces with an inner product defined by the integral shown in my question. Of course, the question could be extended to other cases, too. – fjgg1549 Dec 03 '17 at 10:42
  • By the way, I use the expression "classical Hilbert space" in the sense of Berberian, S.K. "Introduction to Hilbert Space" , 1961, Oxford University Press. – fjgg1549 Dec 03 '17 at 10:54
  • That does not help me understand what "classical Hilbert space" means because I don't have a copy of that book. – Qiaochu Yuan Dec 03 '17 at 18:30
  • (Classical Hilbert space in the sense of Berberian means separable infinite-dimensional complex Hilbert space which as is well-known is isomorphic to $\ell^2$ resp. $L^2$.) – Frederik vom Ende Dec 04 '17 at 16:10

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All Hilbert spaces of the same dimension are isomorphic, no matter how they're defined. Whatever "classical" means, that has nothing to do with it.

Qiaochu Yuan
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