Linear Operators: General theory |
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Page 88
... condition is sufficient . Bonsall [ 1 ] showed that the separa- bility condition cannot be dropped . Ingleton [ 1 ] has given conditions for the Hahn - Banach theorem to hold when the field of scalars is non - Archimedean . ( See also ...
... condition is sufficient . Bonsall [ 1 ] showed that the separa- bility condition cannot be dropped . Ingleton [ 1 ] has given conditions for the Hahn - Banach theorem to hold when the field of scalars is non - Archimedean . ( See also ...
Page 433
... condition implies that K is bounded . For otherwise , there exists an a * € X * such that x * ( K ) is an unbounded ... condition . Further , the condition implies that K is closed , for if x → xq , X2 € K , we set K1 = K co { xn , xn + ...
... condition implies that K is bounded . For otherwise , there exists an a * € X * such that x * ( K ) is an unbounded ... condition . Further , the condition implies that K is closed , for if x → xq , X2 € K , we set K1 = K co { xn , xn + ...
Page 487
... condition is satisfied . Q.E.D. 6. Operators with Closed Range of an It was observed in Lemma 2.8 that the closure ... condition , and define a ( possibly dis- continuous ) linear functional y on Yo = UX by the formula y * ( Ux ) x * ( x ) ...
... condition is satisfied . Q.E.D. 6. Operators with Closed Range of an It was observed in Lemma 2.8 that the closure ... condition , and define a ( possibly dis- continuous ) linear functional y on Yo = UX by the formula y * ( Ux ) x * ( x ) ...
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