Closed Linear Operators on Banach Spaces |
Contents
Dimension of the null space and codimension of the range 7 Closed linear operators with closed range | 7 |
Closed linear operators with nonclosed range | 8 |
CHAPTER | 9 |
12 other sections not shown
Common terms and phrases
AM₁ assume B-¹N B-¹Y 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(T₁ D(Tp denote dense in Lp dense in X difficult to verify dim N(T domain D(T Dp(T dµx dµy exists a sequence extension of Tp f(Tx fe D(T finite number finite perturbation follows from Theorem Furthermore ƒ e D(T graph G(T Hence Hilbert space hypothesis implies kernel L₁ L₂ linear subspace linearly independent M₁ M₂ mapping N₁ non-closed range norm topology null space number of values operator with closed operator with domain preceding lemma Proof proper subspace property P(B proved R(A+B range R(T reflexive satisfying smallest closed linear ST(x subspace spanned T+2S T₁ Theorem 2.4 x₁ Y₁