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I am reading a paper that defines a Gelfand triples. The paper states:

"We define the Gelfand triple of Hilbert spaces $V \subset H \subset V^*$ by

$$H= (L^2(D), <\cdot, \cdot>, ||\cdot||), \quad V= (H_0^1 (D), <\nabla\cdot, \nabla\cdot>, ||\cdot||_V = || \nabla\cdot ||)$$

where $D \subset \mathbb{R}^d$ denotes a bounded open set, with Lipschitz boundary $\partial D$, and $v^*$ is the dual of $V$ with respect to the pairing induced by $H$."

What is $L^2(D)$ and what is $H_0^1(D)$? How would I search for this (on google or here)?

I would guess $L^2(D)$ is functions that are $L^2$ integrable on D, i.e. $\int_D |f|^2 d\mu < \infty$ (though I'm not sure what the measure $\mu$ is in this context). What about $H_0^1$?

EdOverflow
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Mark
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    Say you didn't know anything about the formula being described in the paper, the best advice I can give you would be to try searching using the raw $\LaTeX$ symbols on Google. So for instance, H^1_0 returned results for Sobolev spaces, which should allow you to refine your search even further. If you want to search on a particular site using Google, remember to use site:example.com [search term].

    Another way of solving this problem is to search for other definitions of the same concept online. Instead of searching for the symbols, try looking up Gelfand triple of Hilbert spaces.

     – EdOverflow Jan 09 '19 at 09:22
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    To the good advice of EdOverflow, let me add that, if you find a sketchy definition like the one you wrote, it is probably just a reminder of some concept that the author considers as "well-known". So if you don't know anything about it, it may be a good idea to temporarily close the article you are reading and gather some more information somewhere else. (The hard part is to do this without falling in the rabbit hole). – Giuseppe Negro Jan 09 '19 at 09:45

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