Linear Operators: General theory |
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Page 120
If we put t = al!qb~1,p we obtain the inequality ab 5S a1>jp+bqlq, valid for a, b > 0
; hence, the inequality \ab\ ^ lal'/p + l^l*/? is valid for all scalars a, b. Putting a = /(*
)/!/!.. * = g(*)/|g|*. we find that It follows from Lemma 2.12, Theorem 2.19(a), ...
If we put t = al!qb~1,p we obtain the inequality ab 5S a1>jp+bqlq, valid for a, b > 0
; hence, the inequality \ab\ ^ lal'/p + l^l*/? is valid for all scalars a, b. Putting a = /(*
)/!/!.. * = g(*)/|g|*. we find that It follows from Lemma 2.12, Theorem 2.19(a), ...
Page 121
We observe that the inequality of Minkowski and (in the case of scalar valued
functions) the inequality of Holder may be regarded as applying to the spaces L„.
This observation is obvious in the case of Minkowski's inequality. To see that it is
...
We observe that the inequality of Minkowski and (in the case of scalar valued
functions) the inequality of Holder may be regarded as applying to the spaces L„.
This observation is obvious in the case of Minkowski's inequality. To see that it is
...
Page 248
The above inequality, known as the Schwarz inequality, will be proved first. It
follows from the postulates for that the Schwarz inequality is valid if either x or y is
zero. Hence suppose that X 0 t£ y. For an arbitrary complex number a 0 g (x+cty,
...
The above inequality, known as the Schwarz inequality, will be proved first. It
follows from the postulates for that the Schwarz inequality is valid if either x or y is
zero. Hence suppose that X 0 t£ y. For an arbitrary complex number a 0 g (x+cty,
...
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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 |