Linear Operators: General theory |
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Page 483
If T is weakly compact, then TS is compact in the 9)* topology of SJ) and thus x(
TS) is compact and hence closed in the s3)* topology of?)**. Thus if T is weakly
compact, (i) yields T**(Sj) Q x(TS). According to Theorem V.4.5, S1 = S**, and so
...
If T is weakly compact, then TS is compact in the 9)* topology of SJ) and thus x(
TS) is compact and hence closed in the s3)* topology of?)**. Thus if T is weakly
compact, (i) yields T**(Sj) Q x(TS). According to Theorem V.4.5, S1 = S**, and so
...
Page 494
From IV. 10. 2 we conclude that T* maps the unit sphere of X* into a conditionally
weakly compact set of rca(S), and therefore T* is a weakly compact operator. By
Theorem 4.8 this implies that T is a weakly compact operator. Q.E.D. 4 Theorem.
From IV. 10. 2 we conclude that T* maps the unit sphere of X* into a conditionally
weakly compact set of rca(S), and therefore T* is a weakly compact operator. By
Theorem 4.8 this implies that T is a weakly compact operator. Q.E.D. 4 Theorem.
Page 507
Let (S, Z, u) be a a- finite positive measure space, and let T be a weakly compact
operator on L^S, Z, p) to a separable subset of the B-space 3c. Then there exists
a fi- essentially unique bounded measurable function x(-) on S to a weakly ...
Let (S, Z, u) be a a- finite positive measure space, and let T be a weakly compact
operator on L^S, Z, p) to a separable subset of the B-space 3c. Then there exists
a fi- essentially unique bounded measurable function x(-) on S to a weakly ...
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Contents
Preliminary Concepts | 1 |
B Topological Preliminaries | 10 |
Algebraic Preliminaries 84 | 34 |
Copyright | |
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Common terms and phrases
Acad additive set function algebra Amer analytic arbitrary B-space ba(S Banach spaces Borel sets ca(S Cauchy sequence closed unit sphere compact Hausdorff space compact operator complex numbers contains continuous functions convex set Corollary countably additive Definition denote dense differential equations Doklady Akad Duke Math element equivalent exists finite dimensional function defined function f Hausdorff space Hence Hilbert space homeomorphism implies inequality integral interval isometric isomorphism Lemma linear map linear operator linear topological space LP(S measurable functions measure space metric space Nauk SSSR N. S. neighborhood non-negative normed linear space null set open set operator topology positive measure space Proc properties proved real numbers reflexive Riesz Russian scalar semi-group sequentially compact Show simple functions subset subspace Suppose theory TM(S topological space Trans valued function Vber vector space weak topology weakly compact weakly sequentially compact zero
Bibliographic information
| Title | Linear Operators: General theory Volume 7 of Pure and applied mathematics Volume 1 of Linear Operators, Nelson Dunford |
| Authors | Nelson Dunford, Jacob T. Schwartz |
| Publisher | Interscience Publishers, 1958 |
| Original from | the University of Michigan |
| Digitized | 20 Nov 2007 |
| Length | 2592 pages |
| Subjects | › › Algebra, Universal Algebras, Linear Linear operators Mathematical analysis Mathematics / Algebra / Linear |
| Export Citation | BiBTeX EndNote RefMan |