Linear Operators: General theory |
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Page 253
... orthonormal basis for 5 ; ( iii ) x2 = Σves | ( x , y ) | 2 , x € H. PROOF . The equivalence of statements ( i ) and ... orthonormal bases of a given Hilbert space S have the same cardinality . PROOF . If is finite dimensional , the ...
... orthonormal basis for 5 ; ( iii ) x2 = Σves | ( x , y ) | 2 , x € H. PROOF . The equivalence of statements ( i ) and ... orthonormal bases of a given Hilbert space S have the same cardinality . PROOF . If is finite dimensional , the ...
Page 254
... orthonormal basis of a Hilbert space § is its dimension . 16 THEOREM . Two Hilbert spaces are isometrically isomorphic if and only if they have the same dimension . PROOF . Let U be an isometric isomorphism of 1 onto 2. Then . if x and ...
... orthonormal basis of a Hilbert space § is its dimension . 16 THEOREM . Two Hilbert spaces are isometrically isomorphic if and only if they have the same dimension . PROOF . Let U be an isometric isomorphism of 1 onto 2. Then . if x and ...
Page 346
... orthonormal basis . 45 Let { y } be an orthonormal basis for Hilbert space H. Show that a bounded subset 4 of 5 is conditionally compact if and only if there exists , for each ɛ > 0 , a finite collection yx , ' orthonormal basis { y } ...
... orthonormal basis . 45 Let { y } be an orthonormal basis for Hilbert space H. Show that a bounded subset 4 of 5 is conditionally compact if and only if there exists , for each ɛ > 0 , a finite collection yx , ' orthonormal basis { y } ...
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