Hint: Let . Show that (i) this sum converges (because the sequence (sin 1/n) is square-summable); (ii) ; (iii) each finite partial sum is in Y_1+Y_2.

Let Y_n be the subspace of X spanned by the first n basis vectors. Then T_n permutes these basis vectors cyclically, and it follows that the restriction of T_n to Y_n has norm 1 and is normal. On the orthogonal complement of Y_n, T_n acts as the identity. Hence T_n is bounded (with norm 1) and normal.

Clearly for each basis vector. Since linear combinations of basis vectors are dense in X, and the operators T_n are uniformly bounded, it follows that for all x. [This shows that the set of normal operators is not closed, since S is not normal.]