Product of inner products Is product of innerproduct again a inner product of two vectors? 
For example - Is $ (< u,v >)(< x,y >) = < m,n > $? And if yes is m and n unique and how do we calculate those?  

Of course the product, which is after all a scalar (call it $S$) , can be written as an inner product of two vectors, e.g., $\vec{n} = (S, 0, 0, \ldots), \vec{m} = (1, 0, 0, \ldots)$.  But the decomposition of that scalar is not unique, and in fact, you can find an infinite number of correct $\vec{m} $ cevtors for any specified non-zero $\vec{n}$.
