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Let $H$ a hilbert space with inner product $\left<.,.\right>$. We denote $\|\cdot \|$ the norm induced by the inner product. Lax-miligram tels us that there is a one-to-one correspondance between Continuous and elliptic bilinear form $a:H\times H\to \mathbb R$ and continuous linear functional $L:H\to \mathbb R$. I.e. that if $a$ is continuous (i.e. $a(u,v)\leq K\|u\|\|v\|$), elliptic (i.e. $a(u,u)\geq C\|u\|^2$) and if $L$ is a $L:H\to \mathbb R$ is continuous an linear, then there is a unique $u\in H$ s.t. for all $v\in H$, $$a(u,v)=L(v).$$

Is this a sort of generalization of Riezs representation theorem ? Because it looks very similar, but a bit more general.

user659895
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1 Answers1

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It is the other way around, in my opinion.

One usually applies the Riesz operator (From Riesz' theorem) in your Hilbert space to show Lax-Milgram. Idea:

  • Seeking solution to $Ax = f$, A being strongly positive and bounded.

  • There is a bijective mapping, I, by Riesz.

  • Define $ \phi (x) = x + \tau I(f-Ax)$

  • Show that $\phi$ is a contraction, and Banachs Fixed Point Theorem yields Lax-Milgram.

smallStackBigFlow
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