Linear Operators: General theory |
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Page 36
... linear space , or vector space over a field Þ is an additive group together with an operation m : × X → X , written ... linear combination of the vectors x , y , ... , z . A linear subspace , subspace or linear manifold in a vector ...
... linear space , or vector space over a field Þ is an additive group together with an operation m : × X → X , written ... linear combination of the vectors x , y , ... , z . A linear subspace , subspace or linear manifold in a vector ...
Page 37
... linear transformation on a linear space X is a homomorphism on the additive group X which commutes with the operations of multiplication by scalars . If T : X → Y and U : Y → 3 are linear transformations , and X , Y , Z are linear ...
... linear transformation on a linear space X is a homomorphism on the additive group X which commutes with the operations of multiplication by scalars . If T : X → Y and U : Y → 3 are linear transformations , and X , Y , Z are linear ...
Page 421
... linear space , and let l ' be a total subspace of X. Then the linear functionals on X which are continuous in the I topology are precisely the functionals in T. The proof of Theorem 9 will be based on the following lemma . 10 .... LEMMA ...
... linear space , and let l ' be a total subspace of X. Then the linear functionals on X which are continuous in the I topology are precisely the functionals in T. The proof of Theorem 9 will be based on the following lemma . 10 .... LEMMA ...
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 continuous linear converges convex set Corollary countably additive DEFINITION disjoint Doklady Akad E₁ element exists f₁ finite dimensional function defined function f Hausdorff space Hence Hilbert space integral isometric isomorphism K₁ L₁ L₁(S Lebesgue Lemma Let f linear functional 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 Russian S₁ scalar semi-group sequentially compact Show subset subspace Suppose theory TM(S topological space u-integrable u-measurable uniformly weak topology weakly compact weakly sequentially compact zero ΕΕΣ