(a) Let A = B(X) where X is a Banach space. Suppose there exists m >0 such that ||Ax|| ≥ m||x||, Vx € X. Show that Image A is closed in X. (b) Let A = B(H) be self adjoint, where H is a Hilbert space. Let λ EC such that Imλ 0. Prove that || Ax − Ax|| ≥ |Im\| ||x||, Vx € H. Prove that X is a regular point of A.

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