# A stronger Cauchy-Schwarz in infinite dimensional Hilbert spaces?

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 $$L^2$$?

• 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. – Alexandre Eremenko Jul 25 at 14:26
• Can you demonstrate such an example? – UserA Jul 25 at 14:32
• What example? For $\ell^2$, just replace finite sums with infinite sums. – Alexandre Eremenko Jul 25 at 14:36
• @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 $L^4$). – Jochen Glueck Jul 25 at 14:52
• 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. – Jochen Glueck Jul 25 at 21:52