## Closed Linear Operators on Banach Spaces |

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assume Banach space bounded linear functional bounded linear operator bounded with respect C D(T closed linear extension closed linear operator closed range closed sets closed subspace closure of D(T D(Tm D(TP denoted dense in Lp difficult to verify domain D(T exists a sequence extension of Tp f(Tx fe DP(T feLP fi(T finite number finite perturbation follows from Lemma follows from Theorem Fubini's theorem Furthermore g e Lq geLq graph G(T Hence Hilbert space hypothesis implies kernel linear subspace linearly independent non-closed range norm topology number of values operator with closed operator with domain positive constant preceding lemma Proceeding by induction Proof proper subspace property P(B proved quotient space range R(T reflexive RIDDERKERK satisfying shows smallest closed linear subspace spanned subspaces X0 Theorem 2.4 Tx and B(x tx(A weak x e D(T XB)x