Linear Operators: General theory |
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Page 289
... L is isometrically iso- morphic to L , so that there is a functional y * € L * such that x ** ( x * ) = y * ( g ) when g and x * are connected , as in Theorem 1 , by the formula x * † = √ ̧f ( 8 ) g ( s ) μ ( ds ) , fe Lp . Applying ...
... L is isometrically iso- morphic to L , so that there is a functional y * € L * such that x ** ( x * ) = y * ( g ) when g and x * are connected , as in Theorem 1 , by the formula x * † = √ ̧f ( 8 ) g ( s ) μ ( ds ) , fe Lp . Applying ...
Page 302
... L , which spring from its natural ordering , and which will be useful later . We say that fe L , ( S , Σ , μ ) is positive and write ƒ ≥ 0 if f ( s ) ≥ 0 for μ - almost all 8 S. If f and f2 are ... ( s ) } is 302 IV.8.22 IV . SPECIAL SPACES.
... L , which spring from its natural ordering , and which will be useful later . We say that fe L , ( S , Σ , μ ) is positive and write ƒ ≥ 0 if f ( s ) ≥ 0 for μ - almost all 8 S. If f and f2 are ... ( s ) } is 302 IV.8.22 IV . SPECIAL SPACES.
Page 519
... L , ( S , Σ , μ ) into C ( S ) is a compact operator when it is regarded as mapping into L „ ( S , Σ , μ ) . 57 Let ( S , E , μ ) be a positive finite measure space and let K ( s , t ) be bounded and measurable on SXS . Let T in B ( L1 ...
... L , ( S , Σ , μ ) into C ( S ) is a compact operator when it is regarded as mapping into L „ ( S , Σ , μ ) . 57 Let ( S , E , μ ) be a positive finite measure space and let K ( s , t ) be bounded and measurable on SXS . Let T in B ( L1 ...
Contents
Exercises | 9 |
Definitions and Basic Properties | 13 |
Metric Spaces | 19 |
Copyright | |
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A₁ Acad additive set function algebra Amer analytic arbitrary B-space B₁ ba(S Banach spaces Borel sets Cauchy sequence closed sets compact Hausdorff space compact operator complex numbers contains continuous functions continuous linear converges convex set Corollary countably additive DEFINITION denote dense disjoint Doklady Akad E₁ elements ergodic exists f₁ finite number follows function defined function f Hausdorff space Hence Hilbert space homomorphism 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 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 Theorem theory topological space u-integrable u-measurable u-null uniformly unit sphere vector space weakly compact zero ΕΕΣ