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Real Analysis
book

Real Analysis

by V. Karunakaran
May 2024
Intermediate to advanced content levelIntermediate to advanced
585 pages
15h 38m
English
Pearson India
Content preview from Real Analysis
“real: chapter_14” 2011/5/22 23:43 page3—#3
L
p
-Spaces 14-3
14.3 PROPERTIES OF L
P
-SPACES
Theorem 14.3.1 For 1 p ≤∞, L
p
(R) is a real vector space.
For the proof of the above theorem, we need the following
inequalities.
Theorem 14.3.2 (Holder’s Inequality) For 1 < p < , the conjugate
exponent of p, denoted by q, is defined by
1
p
+
1
q
= 1. If f L
p
(R) and
g L
q
(R), then fg L
1
(R) and ||fg||
1
≤||f ||
p
||g||
q
.
Proof The proof is similar to the proof given in Theorem 10.4.2.
However we shall give the details here for the sake of completeness.
If ||f ||
p
= 0or||g||
q
= 0, f (x) = 0 a.e.org(x) = 0 a.e. In this
case f (x) · g(x) = 0 a.e. and ||fg||
1
= 0 =||f
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Publisher Resources

ISBN: 9781299447561Publisher Website