Linear Operators: General theory |
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Page 36
... vector space are called vectors . The elements of the coefficient field are called scalars . The operations xxx and x → a + x , where a € Ø and a € X , are called scalar multiplication by x , and translation by a , respectively . A sum ...
... vector space are called vectors . The elements of the coefficient field are called scalars . The operations xxx and x → a + x , where a € Ø and a € X , are called scalar multiplication by x , and translation by a , respectively . A sum ...
Page 250
... vector in A has norm one and if every pair of distinct vectors in A are 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 ...
... vector in A has norm one and if every pair of distinct vectors in A are 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 ...
Page 318
... Vector Valued Measures The theorems on spaces of set functions proved in the last section permit us to develop a more satisfactory theory of vector valued count- ably additive set functions ( briefly , vector valued measures ) than we ...
... Vector Valued Measures The theorems on spaces of set functions proved in the last section permit us to develop a more satisfactory theory of vector valued count- ably additive set functions ( briefly , vector valued measures ) than we ...
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