The space V has to be more than a vector space. It must be an inner-product space in order for the adjoint operation to be defined.

For (1), it's clear that if Tx=0 then T*Tx=0. So N(T) ⊆ N(T*T). For the reverse inclusion, notice that .

For (2), suppose T has (finite-dimensional) range , and let be the orthogonal complement of in V. If and then , so that . Therefore The same argument with T and T* interchanged gives the reverse inequality.