Linear Operators: General theory |
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Page 36
... independent set . The cardinality of a Hamel basis is a number independent of the Hamel basis ; it is called the dimension of the linear space . This independence is readily proved if there is a finite Hamel basis , in which case the ...
... independent set . The cardinality of a Hamel basis is a number independent of the Hamel basis ; it is called the dimension of the linear space . This independence is readily proved if there is a finite Hamel basis , in which case the ...
Page 251
... independent of the order in which its terms are arranged . PROOF . For m➤ n m m m m | Σ α ; y ; | 2 = ( Σ α¡Y¿ ‚ Σ α‚ÿ ; ) · Σαν Σ - Σ Σκά i = n i = n j = n = - m = 1/10 Σ Σα , α , ( 1ι , 3 , ) = Σ | αι , i = n j = n in and so the one ...
... independent of the order in which its terms are arranged . PROOF . For m➤ n m m m m | Σ α ; y ; | 2 = ( Σ α¡Y¿ ‚ Σ α‚ÿ ; ) · Σαν Σ - Σ Σκά i = n i = n j = n = - m = 1/10 Σ Σα , α , ( 1ι , 3 , ) = Σ | αι , i = n j = n in and so the one ...
Page 252
... independent of the order of its terms . Now it is clear that E is a linear operator with Ex = x for x in A. Thus Exx for x in the closed linear manifold A , determined by A. Also Ex = 0 if x is orthogonal to 4. Thus E is the orthogonal ...
... independent of the order of its terms . Now it is clear that E is a linear operator with Ex = x for x in A. Thus Exx for x in the closed linear manifold A , determined by A. Also Ex = 0 if x is orthogonal to 4. Thus E is the orthogonal ...
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