Let $H$ be a Hilbert space.
Such that we have a linear operator $T$ $$T: H\rightarrow H'$$ such that there exists $C>0$, $\langle Tx,x\rangle_{H,H'}\geq C \|x\|_{H}^2$.
Obviously, $T$ is injective, but why is $T$ surjective ?
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emacs drives me nuts
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BrianTag
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Perhaps you can use this: https://math.stackexchange.com/q/665235/27978 ? – copper.hat Feb 17 '20 at 17:32
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1How is $\langle Tx, x \rangle$ defined if $x \in H$ and $Tx \in H'$? – Robert Lewis Feb 17 '20 at 17:43
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Or maybe you mean $H'$ is dual to $H$, but since $H$ is self-dual etc. etc. etc . . . – Robert Lewis Feb 17 '20 at 17:44
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I missed the $H,H'$ confusion. – copper.hat Feb 17 '20 at 18:36
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$H'$ is the topologic dual of $H$, and for the duality product $<.>$ is defined as follows: $<Tx,x>=(Tx)(x)$ since $Tx \in H'$ – BrianTag Feb 18 '20 at 07:54