Hilbert space inequality $|\langle x,y\rangle | \leq \epsilon ||x||^2 + C_{\epsilon}||y||^2$

Question

In prelim prep I came across 'given $\epsilon$ there exists $C_{\epsilon}$ such that $|\langle x,y\rangle | \leq \epsilon ||x||^2 + C_{\epsilon}||y||^2$.

It is asserted without proof, so I've tried proving it. I've tried expanding it and applying Cauchy-Schwarz, neither worked. Any ideas? Thanks

Answer 1

$|\langle x,y\rangle|\le\|x\|\|y\|=(\sqrt{2\epsilon}\|x\|)(\frac{\|y\|}{\sqrt{2\epsilon}})\leq\frac{1}{2}(2\epsilon\|x\|^2+\frac{\|y\|^2}{2\epsilon})=\epsilon\|x\|^2+C_{\epsilon}\|y\|^2$

The last inequality is a variant of the geometric inequality $ab\leq\frac{1}{2}(a^2+b^2)$.