Let $H$ be a Hilbert space, $(x,y)$ denote the inner product of the elements $x,y\in H$, $\|x\|$ denote the norm of $x\in H$, and $k>0$.
Do there exist such $x,y\in H$ that $$ (x,y)\geqslant\frac{k}{\|x-y\|^{2}} ? $$
Thank you very much for your help in advance!
Take $e_1,e_2$ to be the orthonormal basis of $\mathrm{span}<x,y>$ and without loss of generality take $x\in \mathrm{span}<e_1>$. Now we can write the coordinates: $x=ae_1$ and $y=be_1+ce_2$ and evaluate $(x,y)=ab$ and $\|x-y\|^2=(x,x)+(y,y)-2(x,y)=a^2+b^2+c^2-2ab$. If we now rewrite the inequality: $(x,y)\|x-y\|^2\geq k$ and we substitute: $$ab(a^2+b^2+c^2-2ab)\geq k$$ we then do some movement and obtain: $$c^2\geq \frac{k+2(ab)^2}{ab}-(a^2+b^2)$$ Which clearly have infinite solutions. When we fix some $a$ and $b$ real we see that we get our lower bound for a positive $c$.