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

4$\begingroup$ 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

1$\begingroup$ Can you demonstrate such an example? $\endgroup$ – UserA Jul 25 at 14:32

2$\begingroup$ What example? For $\ell^2$, just replace finite sums with infinite sums. $\endgroup$ – Alexandre Eremenko Jul 25 at 14:36

3$\begingroup$ @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

3$\begingroup$ 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