Linear Operators: General theory |
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Page 36
... linearly 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 ...
... linearly 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 ...
Page 578
... linearly independent . Suppose that x1 , ... , xn - 1 are linearly independent , but that a , a12 + ··· an + ... + an - 1xn − 1 · Then n = 0 = ( T − λ „ I ) xn = α1 ( λ 1 —λn ) x1 + = α1 ( λ1 — λn ) x1 + ··· + an - 1 ( ^ n - 1 - λn ) ...
... linearly independent . Suppose that x1 , ... , xn - 1 are linearly independent , but that a , a12 + ··· an + ... + an - 1xn − 1 · Then n = 0 = ( T − λ „ I ) xn = α1 ( λ 1 —λn ) x1 + = α1 ( λ1 — λn ) x1 + ··· + an - 1 ( ^ n - 1 - λn ) ...
Page 848
... Linear dimension , ( 91 ) Linear functional , ( 38 ) . ( See also Functional ) Linear manifold , ( 36 ) . ( See also ... Linearly independent , ( 36 ) Liouville theorem , ( 231 ) L , ( S , E , μ ) , 0 < p < 1 , definition , III.9.29 ...
... Linear dimension , ( 91 ) Linear functional , ( 38 ) . ( See also Functional ) Linear manifold , ( 36 ) . ( See also ... Linearly independent , ( 36 ) Liouville theorem , ( 231 ) L , ( S , E , μ ) , 0 < p < 1 , definition , III.9.29 ...
Contents
Preliminary Concepts | 1 |
B Topological Preliminaries | 10 |
Algebraic Preliminaries | 34 |
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A₁ Acad additive set function algebra Amer analytic arbitrary B-space B₁ 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 ergodic exists f₁ finite dimensional function defined function f 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-integrable u-measurable uniformly weak topology weakly compact weakly sequentially compact zero ΕΕΣ