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AB)x assume Banach space bounded linear functional bounded linear operator bounded with respect By(T closed linear extension closed linear operator closed range closed sets closed subspace closure of D(T D(Tº denote dense in Lp difficult to verify dim N(T domain D(T duº exists a sequence extension of Tp f(Tx fe D(T fe Lp fe Yº finite number finite perturbation follows from Lemma follows from Theorem Furthermore graph G(T Hence Hilbert space hypothesis implies kernel linear subspace linearly independent mapping non-closed range null space number of values operator Tº operator with closed operator with domain positive constant preceding lemma Proceeding by induction Proof proper subspace property P(B proved range R(T reflexive ſ T(x satisfying sequence xn ſh D(T smallest closed linear subspace spanned Theorem 2.4 Tº exists Tºf u-measurable weak x e D(T