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Page 240
The space bs is the linear space of all sequences x — {a„} of scalars for which the
norm n \x\ = sup I 2<x*| is finite. 11. The space cs is the linear space of all
sequences x = {a„} for which the series *s convergent. The norm is n \x\ = sup I
2at|.
The space bs is the linear space of all sequences x — {a„} of scalars for which the
norm n \x\ = sup I 2<x*| is finite. 11. The space cs is the linear space of all
sequences x = {a„} for which the series *s convergent. The norm is n \x\ = sup I
2at|.
Page 256
If, however, each of the spaces Sj, . . ., X„ are Hilbert spaces then it will always be
understood, sometimes without explicit mention, that X is the uniquely
determined Hilbert space with scalar product (iv) xn], [yv. . ., = 2 (*<» where (•, -){
is the ...
If, however, each of the spaces Sj, . . ., X„ are Hilbert spaces then it will always be
understood, sometimes without explicit mention, that X is the uniquely
determined Hilbert space with scalar product (iv) xn], [yv. . ., = 2 (*<» where (•, -){
is the ...
Page 323
A scalar valued measurable function / is said to be integrable if there exists a
sequence {/„} of simple functions such that (i) /„(«) converges to f(s) ^-almost
everywhere; (ii) the sequence {J Efn(s)[/(ds)} converges in the norm of £ for each
EeZ.
A scalar valued measurable function / is said to be integrable if there exists a
sequence {/„} of simple functions such that (i) /„(«) converges to f(s) ^-almost
everywhere; (ii) the sequence {J Efn(s)[/(ds)} converges in the norm of £ for each
EeZ.
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Contents
Preliminary Concepts | 1 |
B Topological Preliminaries | 10 |
Algebraic Preliminaries | 34 |
Copyright | |
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a-field Acad additive set function algebra Amer analytic arbitrary B-space ba(S Banach spaces Borel sets ca(S Cauchy sequence closed linear manifold compact operator complex numbers contains continuous functions continuous linear converges convex set Corollary countably additive Definition denote dense differential equations Doklady Akad element equivalent everywhere exists extended real valued extension fi(E finite dimensional finite number function f Hausdorff space Hence Hilbert space homeomorphism inequality integral interval Lebesgue measure Lemma linear functional linear map linear operator linear topological space LP(S measurable function measure space metric space Nauk SSSR N. S. neighborhood non-negative normed linear space null set open set operator topology positive measure space Proc Proof properties proved real numbers Russian scalar semi-group sequentially compact Show simple functions subset subspace Suppose theory topological space uniformly unique v(fi valued function Vber vector valued weakly compact zero