I know that if $x$ is a column vector in $\mathbb R^n$ then the matrix $xx^T$ is positive semidefinite. This is not difficult to prove (see for instance here).
Now let $V$ be a real $n-$dimensional vector space with positive definite inner product $<*,*>$ and $x_k$ $(k=1,2, ... , n)$ $n$ vectors in $V$. Define the matrix $M$ whose elements are $m_{ij}=<x_i,x_j>$.
Is $M$ positive semidefinite? Could you please help me in the proof, if that is true, or to disprove if it is false?).