Linear Operators: General theory |
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Page 165
If / is a /tj-integrable simple function, it is evident that / is also a //-integrable
simple function, and that J" Ef(s)fa(ds) = J ... It follows from (d) and (e) that a
function / e LP(S, Ev fa, X) is also in LP(S, E, ju, X), and that its norm in both
spaces is the ...
If / is a /tj-integrable simple function, it is evident that / is also a //-integrable
simple function, and that J" Ef(s)fa(ds) = J ... It follows from (d) and (e) that a
function / e LP(S, Ev fa, X) is also in LP(S, E, ju, X), and that its norm in both
spaces is the ...
Page 286
discussion of the present section covers the space lp as well as the space Lp(S,E
,fi). 1 Theorem. If 1 <p < oo and l/p + l/q = 1, there is an isometric isomorphism
between L*(S, E, fi) and LV(S, E, /z) in which corresponding vectors x* and g are
...
discussion of the present section covers the space lp as well as the space Lp(S,E
,fi). 1 Theorem. If 1 <p < oo and l/p + l/q = 1, there is an isometric isomorphism
between L*(S, E, fi) and LV(S, E, /z) in which corresponding vectors x* and g are
...
Page 663
11, weakly sequentially compact as a sequence in L^S, E, fj,). The desired
conclusion follows from Corollary 3. Q.E.D. The preceding general results will
now be applied to the case of an operator in LP(S, E, fi) having the special form Tf
— f<p, ...
11, weakly sequentially compact as a sequence in L^S, E, fj,). The desired
conclusion follows from Corollary 3. Q.E.D. The preceding general results will
now be applied to the case of an operator in LP(S, E, fi) having the special form Tf
— f<p, ...
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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