Linear Operators: General theory |
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Page 51
... neighborhood of f ( x ) . If U is a neighborhood of a such that f ( U ) CVf ( y - 1x ) , then Ux - 1y is a neighborhood of y such that ƒ ( Ux ̄1y ) = f ( U ) ƒ ( x − 1y ) ≤ Vƒ ( y ̄1x ) f ( x − 1y ) = V. Therefore , f is con- tinuous ...
... neighborhood of f ( x ) . If U is a neighborhood of a such that f ( U ) CVf ( y - 1x ) , then Ux - 1y is a neighborhood of y such that ƒ ( Ux ̄1y ) = f ( U ) ƒ ( x − 1y ) ≤ Vƒ ( y ̄1x ) f ( x − 1y ) = V. Therefore , f is con- tinuous ...
Page 56
... neighborhood M of 0 such that M - MCG . For every xe X , x / n → 0 , and so xe nM for large n . Thus 00 ∞ X = UnM ... neighborhood of 0. Thus the closure of the image of a neighborhood of the origin contains a neighborhood of the ...
... neighborhood M of 0 such that M - MCG . For every xe X , x / n → 0 , and so xe nM for large n . Thus 00 ∞ X = UnM ... neighborhood of 0. Thus the closure of the image of a neighborhood of the origin contains a neighborhood of the ...
Page 572
... neighborhood of 2. Then , since f ( σ ( T ) ) = 0 , ƒ has a zero of finite order n at 2. Conse- quently , the function g1 , defined by g1 ( § ) = ( 11 — § ) " / ƒ ( § ) , is analytic in a neighborhood of 2. Let e be a function ...
... neighborhood of 2. Then , since f ( σ ( T ) ) = 0 , ƒ has a zero of finite order n at 2. Conse- quently , the function g1 , defined by g1 ( § ) = ( 11 — § ) " / ƒ ( § ) , is analytic in a neighborhood of 2. Let e be a function ...
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