Question on Hilbert space?

Let T be an arbitrary operator on Hilbert space H then there exists n close linear subspaces M1,M2,..,Mn such that





1) {0} C M1 C M2 C ... C Mn = H ( c = contenement )





2) dim Mi = i





3) Each Mi is invariant under T.

Question on Hilbert space?
It appears that you are considering only a finite dimensional hilbert space, dimension n. The trick is to get started, with dimension 1 subspace M1, then get the rest by induction on the dimension of the subspace.





Since H is finite dimensional, the operator T has a nonzero eigenvector v (using the fact that the scalars for Hilbert space are the complex numbers). So let M1 be the 1-dimensional subspace of multiples of the eigenvector v.





Another way to phrase the question is to show that the space H has a basis for which the operator T is a triangular matrix.