# inner product

How much do you know about positive operators? If you know that they are diagonalisable then this result follows fairly easily: There is an orthonormal basis of V with respect to which T has a diagonal matrix D, whose diagonal entries are the eigenvalues of T. The matrix D has an inverse if and only if none of these eigenvalues is 0. That is equivalent to the condition $\langle Tv,v\rangle > 0$ for all v≠0.