Linear Operators: General theory |
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Page 120
PROOF . The function y ( t ) = tP / p + t - 9 / 9 has a positive derivative for t > 1 ,
and a negative derivative for 0 < t < 1 . Hence , its minimum value for t > 0 is 9 ( 1
) = 1 . If we put t = alla b - 1 / p we obtain the inequality ab < app + 6° / q , valid for
a ...
PROOF . The function y ( t ) = tP / p + t - 9 / 9 has a positive derivative for t > 1 ,
and a negative derivative for 0 < t < 1 . Hence , its minimum value for t > 0 is 9 ( 1
) = 1 . If we put t = alla b - 1 / p we obtain the inequality ab < app + 6° / q , valid for
a ...
Page 121
In the sequel the symbol | rather than [ / ] will be used for an element in Lp . We
observe that the inequality of Minkowski and ( in the case of scalar valued
functions ) the inequality of Hölder may be regarded as applying to the spaces Lo
.
In the sequel the symbol | rather than [ / ] will be used for an element in Lp . We
observe that the inequality of Minkowski and ( in the case of scalar valued
functions ) the inequality of Hölder may be regarded as applying to the spaces Lo
.
Page 248
The above inequality , known as the Schwarz inequality , will be proved first . It
follows from the postulates for H that the Schwarz inequality is valid if either x or y
is zero . Hence suppose that + 0 + y . For an arbitrary complex number a 0 = ( x +
...
The above inequality , known as the Schwarz inequality , will be proved first . It
follows from the postulates for H that the Schwarz inequality is valid if either x or y
is zero . Hence suppose that + 0 + y . For an arbitrary complex number a 0 = ( x +
...
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Contents
Preliminary Concepts | 1 |
B Topological Preliminaries | 10 |
Algebraic Preliminaries | 34 |
Copyright | |
31 other sections not shown
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Common terms and phrases
algebra Amer analytic applied arbitrary assumed B-space Banach spaces bounded called clear closed compact operator complex condition Consequently constant contains continuous functions converges convex convex set Corollary countably additive defined DEFINITION denote dense determined differential dimensional disjoint domain element equation equivalent everywhere Exercise exists extension field finite follows formula function defined function f given Hence Hilbert space identity implies inequality integral interval Lebesgue Lemma limit linear functional linear operator linear space Math neighborhood norm operator operator topology problem projection PROOF properties proved range reflexive representation respect satisfies scalar seen semi-group separable sequence set function Show shown statement subset subspace sufficient Suppose Theorem theory topology u-measurable uniform uniformly unique unit sphere valued vector weak weakly compact zero
Bibliographic information
| Title | Linear Operators: General theory Volume 1 of Linear Operators, Jacob T. Schwartz Volume 7 of Pure and applied mathematics |
| Author | Nelson Dunford |
| Publisher | Interscience Publishers, 1958 |
| Original from | the University of California |
| Digitized | 16 Mar 2011 |
| Length | 2592 pages |
| Export Citation | BiBTeX EndNote RefMan |