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

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


It is known that there folds W1,2(U)L2(U)H-1(U)superscriptW12UsuperscriptL2UsuperscriptH1UW^{{1,2}}(U)\subset L^{2}(U)\subset H^{{-1}}(U). This is clear since for every vH1(U)vsuperscriptH1Uv\in H^{1}(U), uH1(U)(u,v)H1usuperscriptH1Unormal-→superscriptsubscriptuvH1u\in H^{1}(U)\rightarrow(u,v)_{H}^{1} is an element of H-1superscriptH1H^{{-1}}. Moreover for every vL2(U)vsuperscriptL2Uv\in L^{2}(U), uH1(U)(u,v)L2usuperscriptH1Unormal-→superscriptsubscriptuvL2u\in H^{1}(U)\rightarrow(u,v)_{L}^{2} is an element of H-1superscriptH1H^{{-1}}. But I also know that H1superscriptH1H^{1} is a hilbert space and therefore it is isomorphic to its dual by riesz theorem. My problem is now how can there be H1(U)L2(U)H-1superscriptH1UsuperscriptL2UsuperscriptH1H^{1}(U)\subset L^{2}(U)\subset H^{{-1}} as well as that H-1superscriptH1H^{{-1}} can be identified with H1(U)superscriptH1UH^{1}(U)?

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To begin, all seperable Hilbert spaces (more generally, all Hilbert spaces of the same dimension) are isomorphic to each other. Thus, H1superscriptH1H^{1}, L2superscriptL2L^{2}, and H-1superscriptH1H^{{-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 H1superscriptH1H^{1} can be expressed as a map u(u,v)H1normal-→usubscriptuvsubscriptH1u\rightarrow(u,v)_{{H_{1}}} for some vH1vsubscriptH1v\in H_{1}. It doesn’t tell us what would happen should we try using the L2superscriptL2L^{2} norm instead of the H1superscriptH1H^{1} norm; thus, this doesn’t contradict your observation that we wind up with the dual being H-1superscriptH1H^{{-1}} if we write it as a map of the form u(u,v)L2normal-→usubscriptuvsuperscriptL2u\rightarrow(u,v)_{{L^{2}}} instead. Indeed, we may define a map M:H1H-1normal-:Mnormal-→superscriptH1superscriptH1M\colon H^{{1}}\to H^{{-1}} by the condition that (u,v)H1=(u,M(v))H-1subscriptuvsuperscriptH1subscriptuMvsuperscriptH1(u,v)_{{H^{{1}}}}=(u,M(v))_{{H^{{-1}}}} for all u,vH1uvsuperscriptH1u,v\in H^{1}. This MMM then works out to be a linear operator which explicitly implements the isomorphicm between H1superscriptH1H^{{1}} and H-1superscriptH1H^{{-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 H1superscriptH1H^{1}, L2superscriptL2L^{2}, and H-1superscriptH1H^{{-1}} norms explicitly in terms of Fourier coefficients, and work out exactly what the statements made above look like in this case.

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

All three are Hilbert spaces of functions. What distinguishes them is the norm which is used to construct the space. In the case of L2superscriptL2L^{2}, the norm of a function is the integral of the square of the function whereas in H1superscriptH1H^{1}, the norm is based on integrating the derivative of the square of the function and in H-1superscriptH1H^{{-1}} upon integrating the square of the antiderivative.

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

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