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

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