A basic question about the spectral theorem for self adjoint compact operator..

Question

In the statement of this theorem which i state from the book of conway that if T is a compact self adjoint operator on a hilbert space then T have countable no of non zero eigen value ln and there exist orthogonal projection Pn onto Ker(T-lnI) such that PnPm=0 if n≠m and T=summation (lnPn).(convergent in operator norm). now i am stuck how to show RanT=orthogonal direct sum of ImPn i.e closure(<{ImPn:n€N}>) ..where means a subspace spanned by X.. I am able to show only one direction that is ImT is a subspace of the orthogonal direct sum of ImPn..but the other inclision i could not..any help will b appreciated..