Linear Operators: General theory |
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Page 169
... Show that there need not exist any set Ee containing A such that v ( u , E ) = 0 . 3 Suppose that S is the interval ( ∞ , ∞ ) , that Σ is the field of finite sums of intervals half open on the left , and that μ is the restriction of ...
... Show that there need not exist any set Ee containing A such that v ( u , E ) = 0 . 3 Suppose that S is the interval ( ∞ , ∞ ) , that Σ is the field of finite sums of intervals half open on the left , and that μ is the restriction of ...
Page 358
... Show that the range of S1 lies in C ( ∞ ) . n 3 Show that S , I strongly in any one of the spaces C ( k ) , k < ∞ , AC , L ,, 1 ≤ p < ∞ if and only if S , ≤K , where S , de- notes the operator norm of S , in the space . Show that ...
... Show that the range of S1 lies in C ( ∞ ) . n 3 Show that S , I strongly in any one of the spaces C ( k ) , k < ∞ , AC , L ,, 1 ≤ p < ∞ if and only if S , ≤K , where S , de- notes the operator norm of S , in the space . Show that ...
Page 360
... Show that for any ƒ in CBV , ( Sf ) ( x ) → f ( x ) uniformly in a . - > 24 Suppose ( i ) , ( ii ) of the last exercise . Show that if f is in BV then ( Sf ) ( x ) → f ( x ) at each point a where ƒ is continuous . 25 Suppose that ( Sf ) ...
... Show that for any ƒ in CBV , ( Sf ) ( x ) → f ( x ) uniformly in a . - > 24 Suppose ( i ) , ( ii ) of the last exercise . Show that if f is in BV then ( Sf ) ( x ) → f ( x ) at each point a where ƒ is continuous . 25 Suppose that ( Sf ) ...
Contents
B Topological Preliminaries | 10 |
Metric Spaces | 23 |
Product Spaces | 31 |
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A₁ Acad additive set function algebra Amer analytic arbitrary B-space B₁ ba(S Banach spaces Borel sets Cauchy sequence compact operator complex numbers contains continuous functions continuous linear converges convex set Corollary countably additive DEFINITION denote dense differential equations Doklady Akad Duke Math E₁ elements ergodic exists extension f₁ function defined function f Hausdorff space Hence Hilbert space homomorphism inequality integral L₁ L₁(S Lebesgue Lemma Let f linear functional linear map linear operator linear topological space measurable functions measure space metric space Nauk SSSR N. S. neighborhood non-negative normed linear space o-field 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 topological space u-integrable u-measurable u-null uniformly unit sphere valued function vector space weakly compact zero ΕΕΣ