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3 votes
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Convergence in $H^{-2}$ of $L^2$-functions with limit in $L^2$

I don't think so. If this was true this would imply boundedness in $L^2$ for your sequence and in the particular case when $f=0$, this would mean that strong convergence in $H^{-2}$ implies ...
Ayman Moussa's user avatar
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6 votes

Can you always extend an isometry of a subset of a Hilbert Space to the whole space?

This is a consequence of the GNS construction for negative definite kernel, see, e.g., Theorem C.2.3 in the Bekka–Harpe–Valette book. In the present case, let $H'$ be the closed affine span of $A$: ...
YCor's user avatar
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