Question

Let H be a separable Hilbert space with basis {e,n}neN and define Pn as the orthogonal projection onto spanfe1,... ,en (a) A sequence of operators Tn e B(H) is said to converge strongly to T if ||Th-T,nh|| converges to 0 for all h E H (note that strong convergence is actually weaker than operator norm convergence-think of this as the difference between pointwise and uniform convergence). Show that, for any T e B(H), the sequence PTP converges strongly to T (b) An operator FE B(H) is said to be of finite rank if rankF= dim(imF) is finite. If rankF = 1, show that F(h) = (h, g)k for some g, k E H (HINT: Riesz representation theorem) (c) Show that the linear span of rank 1 operators is strongly dense in B(H).

Could someone help me with part (b) and part (c)? Please make the solution clear to understand, thanks!