Linear Operators: General theory |
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Page 249
... closed linear manifold M in ɓ is a closed linear manifold complementary to M , i.e. H = M ↔ N. PROOF . It follows from the linearity and the continuity of the scalar product ( Theorem 1 ) that the orthocomplement of any set M is a closed ...
... closed linear manifold M in ɓ is a closed linear manifold complementary to M , i.e. H = M ↔ N. PROOF . It follows from the linearity and the continuity of the scalar product ( Theorem 1 ) that the orthocomplement of any set M is a closed ...
Page 252
... linear operator with Ex = x for x in A. Thus Ex = x for x in the closed linear manifold A1 determined by A. Also Ex = 0 if x is orthogonal to A. Thus E is the orthogonal projec- tion on A1 . Q.E.D. 11 DEFINITION . A set A is called an ...
... linear operator with Ex = x for x in A. Thus Ex = x for x in the closed linear manifold A1 determined by A. Also Ex = 0 if x is orthogonal to A. Thus E is the orthogonal projec- tion on A1 . Q.E.D. 11 DEFINITION . A set A is called an ...
Page 482
... manifolds . It should be recalled ( IV.4.4 ) that every closed linear manifold in S determines uniquely an ortho - complement ; hence every closed linear manifold X in § determines uniquely a self - adjoint projection E in H with ES = X ...
... manifolds . It should be recalled ( IV.4.4 ) that every closed linear manifold in S determines uniquely an ortho - complement ; hence every closed linear manifold X in § determines uniquely a self - adjoint projection E in H with ES = X ...
Contents
A Settheoretic Preliminaries | 1 |
10 | 30 |
Three Basic Principles of Linear Analysis | 49 |
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A₁ additive set function algebra analytic arbitrary B-space B₁ ba(S Banach Borel sets ca(S Cauchy sequence closed unit sphere compact Hausdorff space compact operator complex numbers conditionally compact contains continuous functions convex set Corollary countably additive DEFINITION denote dense E₁ element equation equivalent exists f₁ finite dimensional function defined function f Hausdorff space Hence Hilbert space homeomorphism implies inequality integral isometric isomorphism K₁ L₁ L₁(S Lebesgue measure Lemma Let f linear map linear operator linear topological space Math measurable functions measure space metric space neighborhood non-negative non-zero normed linear space o-field o-finite open set operator topology positive measure space properties proved real numbers reflexive Riesz S₁ scalar semi-group sequentially compact Show subset subspace Suppose theory TM(S topological space u-integrable u-measurable uniformly valued function weak topology weakly compact weakly sequentially compact X₁ zero ΕΕΣ