Linear Operators: General theory |
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Page 32
... formal definition follows : χα 1 DEFINITION . Let ( X , 7 ) be an indexed family of topological spaces . Then the product topology is the topology 7 in X = P ̧X ̧ obtained by taking the collection of all sets U PU , where each U is open ...
... formal definition follows : χα 1 DEFINITION . Let ( X , 7 ) be an indexed family of topological spaces . Then the product topology is the topology 7 in X = P ̧X ̧ obtained by taking the collection of all sets U PU , where each U is open ...
Page 90
... formal sums Σay , with suitable identifications . Such a space is called the direct product ( although the terms tensor , cross , and Kronecker product are also used ) . If X and Y are B - spaces , it is desired to define a topology on ...
... formal sums Σay , with suitable identifications . Such a space is called the direct product ( although the terms tensor , cross , and Kronecker product are also used ) . If X and Y are B - spaces , it is desired to define a topology on ...
Page 359
... formal series 00 ( 27 ) -1 / 2Σfeinx ∞- is called the Fourier series of f . ∞ 13 Show that if { c } is a sequence with n2 < ∞o , then there is some f in L2 whose nth Fourier coefficient is c . Show that , conversely , if f is in L2 ...
... formal series 00 ( 27 ) -1 / 2Σfeinx ∞- is called the Fourier series of f . ∞ 13 Show that if { c } is a sequence with n2 < ∞o , then there is some f in L2 whose nth Fourier coefficient is c . Show that , conversely , if f is in L2 ...
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