Linear Operators: General theory |
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Page 20
... sequence { a } is said to be convergent if an → a for some a . A sequence { a } in a metric space is a Cauchy sequence if limm , n ( am , an ) = 0. If every Cauchy se- quence is convergent , a metric space is said to be complete . The ...
... sequence { a } is said to be convergent if an → a for some a . A sequence { a } in a metric space is a Cauchy sequence if limm , n ( am , an ) = 0. If every Cauchy se- quence is convergent , a metric space is said to be complete . The ...
Page 68
... Cauchy sequence of scalars for each * * is called a weak Cauchy sequence . The space X is said to be weakly complete if every weak Cauchy sequence has a weak limit . In Chapter V , a topology is introduced in certain linear spaces in ...
... Cauchy sequence of scalars for each * * is called a weak Cauchy sequence . The space X is said to be weakly complete if every weak Cauchy sequence has a weak limit . In Chapter V , a topology is introduced in certain linear spaces in ...
Page 122
... sequence of functions in L ( S , E , μ , X ) and let f be a function on S to X. Then f is in L , and fnf converges ... Cauchy sequence in L1 ( S ) . For ε > 0 there is , by ( iii ) , a set E with v ( μ , E ) < ∞ such that --- \ 8n − 8m ...
... sequence of functions in L ( S , E , μ , X ) and let f be a function on S to X. Then f is in L , and fnf converges ... Cauchy sequence in L1 ( S ) . For ε > 0 there is , by ( iii ) , a set E with v ( μ , E ) < ∞ such that --- \ 8n − 8m ...
Contents
A Settheoretic Preliminaries | 1 |
10 | 30 |
Three Basic Principles of Linear Analysis | 49 |
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A₁ additive set function algebra analytic arbitrary B-space B₁ ba(S Banach Borel sets ca(S Cauchy sequence closed unit sphere compact Hausdorff space compact operator complex numbers conditionally compact contains continuous functions convex set Corollary countably additive DEFINITION denote dense E₁ element equation equivalent exists f₁ finite dimensional function defined function f Hausdorff space Hence Hilbert space homeomorphism implies inequality integral isometric isomorphism K₁ L₁ L₁(S Lebesgue measure Lemma Let f linear map linear operator linear topological space Math measurable functions measure space metric space neighborhood non-negative non-zero normed linear space o-field o-finite open set operator topology positive measure space properties proved real numbers reflexive Riesz S₁ scalar semi-group sequentially compact Show subset subspace Suppose theory TM(S topological space u-integrable u-measurable uniformly valued function weak topology weakly compact weakly sequentially compact X₁ zero ΕΕΣ