I just read about the notes of functional analysis from my teacher. It gives a definition on cyclic subspace and invariant subspace as following: Let H be a Hilbert space, L is a closed subspace, T is a self-adjoint operator. We say L is an invariant subspace of T if for any $z\in C/R$, $(T-z)^{-1}L\subset L$. We say that L is a cyclic subspace if there exists v such that L is the smallest invariant subspace containing v.
The definition I used in linear algebra is that "L is an invariant subspace if $TL\subset L$" and "L is a cyclic subspace if $L=<v,Tv,T^2v,...T^nv,...>$". How can I prove that these two definitions are equivalent?
Thanks for any help you offer.