The Dualspace of H^1= W^(1,2)

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# The Dualspace of H^1= W^(1,2)

Submitted by tba on Mon, 02/25/2013 - 14:15

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Hi,

It is known that there folds $W^{{1,2}}(U)\subset L^{2}(U)\subset H^{{-1}}(U)$. This is clear since for every $v\in H^{1}(U)$, $u\in H^{1}(U)\rightarrow(u,v)_{H}^{1}$ is an element of $H^{{-1}}$. Moreover for every $v\in L^{2}(U)$, $u\in H^{1}(U)\rightarrow(u,v)_{L}^{2}$ is an element of $H^{{-1}}$. But I also know that $H^{1}$ is a hilbert space and therefore it is isomorphic to its dual by riesz theorem. My problem is now how can there be $H^{1}(U)\subset L^{2}(U)\subset H^{{-1}}$ as well as that $H^{{-1}}$ can be identified with $H^{1}(U)$?

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## Re: The Dualspace of H^1= W^(1,2)

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## Re: The Dualspace of H^1= W^(1,2)

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## Re: The Dualspace of H^1= W^(1,2)

To begin, all seperable Hilbert spaces (more generally, all Hilbert spaces of the same dimension) are isomorphic to each other. Thus, $H^{1}$, $L^{2}$, and $H^{{-1}}$ are all isomorphic to each other as Hilbert spaces. In this respect, there is no difficulty.

As for the specific form of the dual, all that Riesz’ theorem tells us is that every bounded functional on $H^{1}$ can be expressed as a map $u\rightarrow(u,v)_{{H_{1}}}$ for some $v\in H_{1}$. It doesn’t tell us what would happen should we try using the $L^{2}$ norm instead of the $H^{1}$ norm; thus, this doesn’t contradict your observation that we wind up with the dual being $H^{{-1}}$ if we write it as a map of the form $u\rightarrow(u,v)_{{L^{2}}}$ instead. Indeed, we may define a map $M\colon H^{{1}}\to H^{{-1}}$ by the condition that $(u,v)_{{H^{{1}}}}=(u,M(v))_{{H^{{-1}}}}$ for all $u,v\in H^{1}$. This $M$ then works out to be a linear operator which explicitly implements the isomorphicm between $H^{{1}}$ and $H^{{-1}}$ discussed above.

A nice little exercise for seeing what is going on here in concrete terms is to go to the case of functions on an interval, write down the $H^{1}$, $L^{2}$, and $H^{{-1}}$ norms explicitly in terms of Fourier coefficients, and work out exactly what the statements made above look like in this case.

## hilbert space

I am a little rusty. What is the difference between H1, L2, and H-1 ?

## re: hilbert spaces

All three are Hilbert spaces of functions. What distinguishes them is the norm which is used to construct the space. In the case of $L^{2}$, the norm of a function is the integral of the square of the function whereas in $H^{1}$, the norm is based on integrating the derivative of the square of the function and in $H^{{-1}}$ upon integrating the square of the antiderivative.

## re: Hilbert spaces

It looks to me that these would be different spaces in that they are made up of different functions.