Linear Operators: General theory |
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Page 358
... strongly in L1 if and only if S → I strongly in C. 5 Show that S , → I strongly in L , if and only if S1 → I strongly in L. ( 1 / p + 1 / q = 1 ) . 6 Show that ( Sf ) ( x ) converges to f ( x ) uniformly on [ 0 , 27 ] for all fe C ...
... strongly in L1 if and only if S → I strongly in C. 5 Show that S , → I strongly in L , if and only if S1 → I strongly in L. ( 1 / p + 1 / q = 1 ) . 6 Show that ( Sf ) ( x ) converges to f ( x ) uniformly on [ 0 , 27 ] for all fe C ...
Page 472
... strongly differentiable at the point a . Banach [ 1 ; p . 168 ] showed that the norm in C [ 0 , 1 ] is strongly differentiable at xo e C [ 0 , 1 ] if and only if the function to achieves its maximum at exactly one point . Mazur [ 1 ; p ...
... strongly differentiable at the point a . Banach [ 1 ; p . 168 ] showed that the norm in C [ 0 , 1 ] is strongly differentiable at xo e C [ 0 , 1 ] if and only if the function to achieves its maximum at exactly one point . Mazur [ 1 ; p ...
Page 689
... strongly convergent if they are bounded and if T ( n ) / n converges to zero strongly . PROOF . This follows from Theorem 1 and Theorem II.3.28 . Q.E.D. 4 COROLLARY . Let ( S , E , μ ) be a finite positive measure space and { T ( t ) ...
... strongly convergent if they are bounded and if T ( n ) / n converges to zero strongly . PROOF . This follows from Theorem 1 and Theorem II.3.28 . Q.E.D. 4 COROLLARY . Let ( S , E , μ ) be a finite positive measure space and { T ( t ) ...
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