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In this MSE and question and this MO question, stronger variants of the classical Cauchy-Schwarz inequality have been suggested in finite dimensional spaces. Can we find similar results for infinite dimensional Hilbert spaces such as $L^2$?

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    $\begingroup$ This inequality makes sense only of your vector spaces are real. For real $\ell^2$ or real $L^2$ similar inequalities can be obtained by passing to the limit in finite dimensional inequalities. $\endgroup$ – Alexandre Eremenko Jul 25 at 14:26
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    $\begingroup$ Can you demonstrate such an example? $\endgroup$ – UserA Jul 25 at 14:32
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    $\begingroup$ What example? For $\ell^2$, just replace finite sums with infinite sums. $\endgroup$ – Alexandre Eremenko Jul 25 at 14:36
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    $\begingroup$ @AlexandreEremenko: I agree that it's very simple on $\ell^2$. For $L^2$-spaces, though, the question seems to be much more subtle (starting with the fact that $L^2$ is not contained in $L^4$). $\endgroup$ – Jochen Glueck Jul 25 at 14:52
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    $\begingroup$ Would the person who voted to close as "not about research level mathematics" kindly explain their objections? When the post in the second link was asked and answered, I thought for a while about the $L^2$-case, too, but I did not even come close to any satisfactory answer. So I do not find this question simple or elementary by any means. $\endgroup$ – Jochen Glueck Jul 25 at 21:52

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