Linear Operators: General theory |
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Page 136
... functional with || ƒ || = 1 . ᄆ 3. Let X = ( C [ a , b ] , || . || ∞ ) be the Banach space ( Example 2.2.7 ) . Take ... functional f : X → R by f10 ( x ) = x ( to ) , xEC [ a , b ] . Then , fi is a bounded linear functional with || f1 ...
... functional with || ƒ || = 1 . ᄆ 3. Let X = ( C [ a , b ] , || . || ∞ ) be the Banach space ( Example 2.2.7 ) . Take ... functional f : X → R by f10 ( x ) = x ( to ) , xEC [ a , b ] . Then , fi is a bounded linear functional with || f1 ...
Page 148
... functional F on a vector x will be denoted Fx or ( F , x ) . In some contexts , the letter notation is especially useful showing the duality between x and F ( see below ) . Let us recall that boundedness of a linear functional F means ...
... functional F on a vector x will be denoted Fx or ( F , x ) . In some contexts , the letter notation is especially useful showing the duality between x and F ( see below ) . Let us recall that boundedness of a linear functional F means ...
Page 69
Tailen Hsing, Randall Eubank. LINEAR OPERATOR AND FUNCTIONALS 69 Corollary 3.2.7. If f is a bounded linear functional on a subspace M of a normed vector space X, there is an extension / off defined on all of X with the same norm as 6 on ...
Tailen Hsing, Randall Eubank. LINEAR OPERATOR AND FUNCTIONALS 69 Corollary 3.2.7. If f is a bounded linear functional on a subspace M of a normed vector space X, there is an extension / off defined on all of X with the same norm as 6 on ...
Contents
A Settheoretic Preliminaries | 1 |
B Topological Preliminaries | 10 |
Algebraic Preliminaries | 34 |
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A₁ Acad additive set function algebra Amer analytic arbitrary B-space B₁ ba(S Banach spaces Borel sets Cauchy sequence compact operator complex numbers contains continuous functions continuous linear converges convex set Corollary countably additive DEFINITION denote dense differential equations disjoint Doklady Akad domain E₁ element exists f₁ finite dimensional finite number function defined function f Hausdorff space Hence Hilbert space homeomorphism inequality integral L₁ L₁(S Lebesgue Lemma Let f linear functional linear map linear operator linear topological space measurable function measure space metric space Nauk SSSR N. S. neighborhood non-negative o-field open set operator topology positive measure space Proc PROOF proved real numbers Riesz Russian S₁ scalar semi-group sequentially compact Show spectral strong operator topology subset subspace Suppose T₁ theory topological space u-integrable u-measurable uniformly unit sphere valued function weakly compact zero ΕΕΣ