Suppose $a_1,a_2,...,a_n,b_1,b_2,...,b_n$ are real numbers. Does there exist a constant $K$ (depending on $n$ maybe, but not on $a_i$ or $b_i$) such that $(a_1^2+a_2^2+...+a_n^2)(b_1^2+b_2^2+...+b_n^2)\leq K(a_1b_1+a_2b_2+...+a_nb_n)^2$?
I don't know how to proceed about it. Any approach/idea is appreciated.
I am basically trying to see whether the well-known Cauchy Schwarz inequality can be reversed in some way. Actually, I am interested in the following problem: does a bound on $|u'v|$ imply any bound on $||u||.||v||$? Here $u,v$ are vectors. This is equivalent to the one I mentioned.