Linear Operators: General theory |
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Page 114
... Statement ( b ) is clear from the definitions . Statement ( c ) is evident for u - integrable simple functions ; to prove it in the general case , let f2 be a sequence of finitely valued u - simple functions which determine g . Then the ...
... Statement ( b ) is clear from the definitions . Statement ( c ) is evident for u - integrable simple functions ; to prove it in the general case , let f2 be a sequence of finitely valued u - simple functions which determine g . Then the ...
Page 447
... Statement ( b ) follows from the inequality a 2 2f ( x + ( Y1 + Y2 ) ) ≤ x ( x + ay1 ) + { ( x + ay2 ) . Statement ( c ) is trivial . Statement ( d ) follows from the inequality T ( x , y ) + T ( x , y ) ≥ t ( x , 0 ) = 0 · T ( x , y ) ...
... Statement ( b ) follows from the inequality a 2 2f ( x + ( Y1 + Y2 ) ) ≤ x ( x + ay1 ) + { ( x + ay2 ) . Statement ( c ) is trivial . Statement ( d ) follows from the inequality T ( x , y ) + T ( x , y ) ≥ t ( x , 0 ) = 0 · T ( x , y ) ...
Page 487
... statement that T ( S ) is an equicontinuous subset of C ( S * ) . It follows from Theorem IV.6.7 , that T ( S ) is ... statement is , in general , false . This will be seen in an exercise . However the dual statement is true if the range ...
... statement that T ( S ) is an equicontinuous subset of C ( S * ) . It follows from Theorem IV.6.7 , that T ( S ) is ... statement is , in general , false . This will be seen in an exercise . However the dual statement is true if the range ...
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