Linear Operators: General theory |
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Page 494
... operator T , defined by ( b ) , is a bounded linear operator on C ( S ) to X whose adjoint T * is given by ( d ) . From IV.10.2 we conclude that T * maps the unit sphere of X * into a conditionally weakly compact set of rca ( S ) , and ...
... operator T , defined by ( b ) , is a bounded linear operator on C ( S ) to X whose adjoint T * is given by ( d ) . From IV.10.2 we conclude that T * maps the unit sphere of X * into a conditionally weakly compact set of rca ( S ) , and ...
Page 503
... bounded measurable functions it is bounded and measurable . Moreover , if f≤1 , then | x * Tƒ − { ̧x * x ( t ) f ( t ) μ ( dt ) | - [ s ≤ \ x * Tj − x * Tf \ + { s \ x * x ( t ) − x , x ( t ) || f ( t ) \ μ ( dt ) 28 , which proves ...
... bounded measurable functions it is bounded and measurable . Moreover , if f≤1 , then | x * Tƒ − { ̧x * x ( t ) f ( t ) μ ( dt ) | - [ s ≤ \ x * Tj − x * Tf \ + { s \ x * x ( t ) − x , x ( t ) || f ( t ) \ μ ( dt ) 28 , which proves ...
Page 554
... bounded operator T : Xo Yr has an extension ( with the same < norm ) T : X → Yr if and only if there is a projection ( of norm 1 ) from X onto X. Thus Kakutani [ 6 ] proved that X is a Hilbert space if and only if an arbitrary bounded ...
... bounded operator T : Xo Yr has an extension ( with the same < norm ) T : X → Yr if and only if there is a projection ( of norm 1 ) from X onto X. Thus Kakutani [ 6 ] proved that X is a Hilbert space if and only if an arbitrary bounded ...
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