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Hi. I'm studying Finite Elements Method, I was readding a paper written by Danielle Boffi and in a part dedicated to the approximation of eigenvalues in mixed form, it's about approximating eigenvalues in the Hilbert Spaces [tex]\Phi[/tex] and [tex]\Xi[/tex]

Then it says:

"If we suppose that there exists Hilbert Spaces [tex]H_{\Phi}[/tex] and [tex]H_{\Xi}[/tex] such that the following dense and continuous embeddings hold in a compatible way

[tex]\Phi \subset H_{\Phi} \simeq H_{\Phi} ' \subset \Phi'[/tex]

[tex]\Xi \subset H_{\Xi} \simeq H_{\Xi} ' \subset \Xi'[/tex]

Here [tex]\subset[/tex] means dense and continously embedded.

I'm not sure why it does that, I have read more than one time that they work with a space identified with it's dual space, as they're Hilbert Spaces I know you can do it using Riesz Representation theorem, but I don't exacly see why they're doing this.

I didn't know why they don't simply identify [tex]\Phi[/tex] with [tex]\Phi'[/tex].

I have found an advice against identifying a space which is not [tex]L^{2}[/tex] with it's dual, because otherwise in constructions like this, if [tex]H_{\Phi}=L^{2}[/tex] and [tex]\Phi=H^{1}[/tex] we would end up identifying the four spaces [tex]\Phi \equiv H_{\Phi} \equiv H_{\Phi} ' \equiv \Phi'[/tex] and it would mean that you're identifying a function with it's laplacian which is (and here I'm quoting a book) ""the beggining of the end""

I'm new in this of variational formulations and I don't have a strong background on partial differential equations, I have more background on functional and real analysis, I'm studying this subject to make my Licenciatura's thesis in this but I'm really lost sometimes, this is as clear as I could make the question so feel free to ask me to clarify something if I wasn't clear enough.

PD: I don't know how to write in latex without making a new line.

Then it says:

"If we suppose that there exists Hilbert Spaces [tex]H_{\Phi}[/tex] and [tex]H_{\Xi}[/tex] such that the following dense and continuous embeddings hold in a compatible way

[tex]\Phi \subset H_{\Phi} \simeq H_{\Phi} ' \subset \Phi'[/tex]

[tex]\Xi \subset H_{\Xi} \simeq H_{\Xi} ' \subset \Xi'[/tex]

Here [tex]\subset[/tex] means dense and continously embedded.

I'm not sure why it does that, I have read more than one time that they work with a space identified with it's dual space, as they're Hilbert Spaces I know you can do it using Riesz Representation theorem, but I don't exacly see why they're doing this.

I didn't know why they don't simply identify [tex]\Phi[/tex] with [tex]\Phi'[/tex].

I have found an advice against identifying a space which is not [tex]L^{2}[/tex] with it's dual, because otherwise in constructions like this, if [tex]H_{\Phi}=L^{2}[/tex] and [tex]\Phi=H^{1}[/tex] we would end up identifying the four spaces [tex]\Phi \equiv H_{\Phi} \equiv H_{\Phi} ' \equiv \Phi'[/tex] and it would mean that you're identifying a function with it's laplacian which is (and here I'm quoting a book) ""the beggining of the end""

I'm new in this of variational formulations and I don't have a strong background on partial differential equations, I have more background on functional and real analysis, I'm studying this subject to make my Licenciatura's thesis in this but I'm really lost sometimes, this is as clear as I could make the question so feel free to ask me to clarify something if I wasn't clear enough.

PD: I don't know how to write in latex without making a new line.

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