Compact Linear Operators

Proof. Since the dimensions of R(A) is always small or equal to the dimension of N (A)⊥, X and R(A), and N (A)⊥ are all infinite dimensional. Hence, we can find a sequence (xn) with xn ∈ N (A)⊥ with ‖xn‖ = 1 and 〈xn, xm〉 = 0 for n 6= m. Since A is compact, the sequence (yn) = (Axn) has to contain a convergent subsequence. Thus, for any δ > 0 we can find k and l such that ‖yk − yl‖ < δ. However, ‖A(yk − yl)‖ = ‖xk − xl‖ = ‖xk‖ + ‖xl‖ − 2〈xk, xl〉 = 2, although A†0 = 0. Hence, A† is not continuous.