Linear Operators: General theory |
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Page 122
... fi , fe L , 81 , 82 € Lo , and fi - f2 and g1 - g2 are μ - null functions , then fig1 - f2g2 is a μ - null function ... v ( u , ds ) v ( μ , E ) → 0 E - 0 uniformly in n ; ( iii ) for each ε > 0 there is a set E in Σ with v ( μ , E ̧ ) ...
... fi , fe L , 81 , 82 € Lo , and fi - f2 and g1 - g2 are μ - null functions , then fig1 - f2g2 is a μ - null function ... v ( u , ds ) v ( μ , E ) → 0 E - 0 uniformly in n ; ( iii ) for each ε > 0 there is a set E in Σ with v ( μ , E ̧ ) ...
Page 262
... v ( u , S ) << ∞ , f is μ - integrable . Since the integral ssf ( s ) μ ( ds ) satisfies the in- equality | [ çf ... fi ( s ) = 0 ifs & G and f ( s ) = 1 if se C. Let a1 , ... , a , be complex constants of modulus one such that au ( E ; ) | ...
... v ( u , S ) << ∞ , f is μ - integrable . Since the integral ssf ( s ) μ ( ds ) satisfies the in- equality | [ çf ... fi ( s ) = 0 ifs & G and f ( s ) = 1 if se C. Let a1 , ... , a , be complex constants of modulus one such that au ( E ; ) | ...
Page 267
Nelson Dunford, Jacob T. Schwartz. \ fi ( s ) —fi ( t ) < ε / 3 , te N , i = 1 , · Then , for each f in K , each t in ... V of the unit in G such that ( i ) \ f ( t ) —f ( s ) | < ɛ / 2 , fe K , 8 if t is a point in S with t e V ̧s ...
Nelson Dunford, Jacob T. Schwartz. \ fi ( s ) —fi ( t ) < ε / 3 , te N , i = 1 , · Then , for each f in K , each t in ... V of the unit in G such that ( i ) \ f ( t ) —f ( s ) | < ɛ / 2 , fe K , 8 if t is a point in S with t e V ̧s ...
Contents
A Settheoretic Preliminaries | 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 Cauchy sequence compact operator complex numbers contains continuous functions continuous linear converges convex set Corollary countably additive DEFINITION denote dense differential equations disjoint Doklady Akad domain E₁ element exists f₁ finite dimensional finite number function defined function f Hausdorff space Hence Hilbert space homeomorphism inequality integral L₁ L₁(S Lebesgue Lemma Let f linear functional linear map linear operator linear topological space measurable function measure space metric space Nauk SSSR N. S. neighborhood non-negative o-field open set operator topology positive measure space Proc PROOF proved real numbers Riesz Russian S₁ scalar semi-group sequentially compact Show spectral strong operator topology subset subspace Suppose T₁ theory topological space u-integrable u-measurable uniformly unit sphere valued function weakly compact zero ΕΕΣ