Linear Operators: General theory |
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Page 54
... bounded sets into bounded sets . PROOF . Let X , Y be F - spaces , let T : X - > y be linear and contin- uous , and let BCX be bounded . For every neighborhood V of the zero in Y , there is a neighborhood U of zero in X such that T ( U ) ...
... bounded sets into bounded sets . PROOF . Let X , Y be F - spaces , let T : X - > y be linear and contin- uous , and let BCX be bounded . For every neighborhood V of the zero in Y , there is a neighborhood U of zero in X such that T ( U ) ...
Page 345
... bounded and the set { f ( P ) } , fe A , is quasi - equicontinuous . 37 Let D be a bounded domain . Show that a sequence of func- tions in 4 ( D ) is a weak Cauchy sequence ( a sequence converging weakly to ƒ in A ( D ) ) if and only if ...
... bounded and the set { f ( P ) } , fe A , is quasi - equicontinuous . 37 Let D be a bounded domain . Show that a sequence of func- tions in 4 ( D ) is a weak Cauchy sequence ( a sequence converging weakly to ƒ in A ( D ) ) if and only if ...
Page 436
... bounded ( II.1.7 ) set , then X must be finite dimensional . 6 Let X be a B - space , and X , a subspace of X. Show ... bounded if and only if f ( A ) is a bounded set of scalars for each ƒ € I. 8 Let X be a B - space . Show that a ...
... bounded ( II.1.7 ) set , then X must be finite dimensional . 6 Let X be a B - space , and X , a subspace of X. Show ... bounded if and only if f ( A ) is a bounded set of scalars for each ƒ € I. 8 Let X be a B - space . Show that a ...
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