Linear Operators: General theory |
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Page 249
... orthogonal if ( x , y ) = 0. Two manifolds M , N in are orthogonal manifolds if ( M , N ) = 0. We write xy to indicate that x and y are orthogonal , and MN to indicate that M and N are orthogonal . The orthocom- plement of a set ACH is ...
... orthogonal if ( x , y ) = 0. Two manifolds M , N in are orthogonal manifolds if ( M , N ) = 0. We write xy to indicate that x and y are orthogonal , and MN to indicate that M and N are orthogonal . The orthocom- plement of a set ACH is ...
Page 250
... orthogonal . An orthonormal set is said to be complete if no non - zero vector is orthogonal to every vector in the set , i.e. , A is complete if { 0 } = A. We recall that a projection is a linear operator E with E2 E. A projection E in ...
... orthogonal . An orthonormal set is said to be complete if no non - zero vector is orthogonal to every vector in the set , i.e. , A is complete if { 0 } = A. We recall that a projection is a linear operator E with E2 E. A projection E in ...
Page 251
... orthogonal projection . It is the uniquely determined orthogonal projection with ES = M. For if D is an orthogonal projection with DH = M then ED = D and , since ( I – D ) CH → M , we see that E ( I - D ) 0. Thus = = D = ED + E ( I – D ) ...
... orthogonal projection . It is the uniquely determined orthogonal projection with ES = M. For if D is an orthogonal projection with DH = M then ED = D and , since ( I – D ) CH → M , we see that E ( I - D ) 0. Thus = = D = ED + E ( I – D ) ...
Contents
A Settheoretic Preliminaries | 1 |
B Topological Preliminaries | 10 |
Algebraic Preliminaries | 34 |
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A₁ Acad additive set function algebra Amer analytic arbitrary B-space ba(S Banach spaces Borel sets ca(S Cauchy sequence compact Hausdorff space compact operator complex numbers contains continuous functions converges convex set Corollary countably additive DEFINITION dense disjoint Doklady Akad E₁ element exists f₁ finite dimensional function defined function f g₁ Hausdorff space Hence Hilbert space homeomorphism implies inequality integral isometric isomorphism K₁ L₁ L₁(S Lebesgue Lemma Let f linear manifold linear map linear operator linear topological space measurable functions measure space metric space Nauk SSSR N. S. neighborhood non-negative o-field o-finite open set operator topology positive measure space Proc PROOF properties proved real numbers reflexive Riesz S₁ scalar semi-group sequentially compact Show subset subspace Suppose theory topological space u-measurable uniformly weak topology weakly compact weakly sequentially compact zero ΕΕΣ