
“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