Page 30 - INTRODUCTION TO THE CALCULUS OF VARIATIONS

p
L spaces 17

p
p
0
(ii) Hölder inequality asserts that if u ∈ L (Ω) and v ∈ L (Ω) where
1
0
0
1/p +1/p =1 (i.e., p = p/ (p − 1))and 1 ≤ p ≤∞ then uv ∈ L (Ω) and
moreover
kuvk 1 ≤ kuk p kvk p 0 .
L L L
In the case p =2 and hence p =2, Hölder inequality is nothing else than
0
Cauchy-Schwarz inequality
Z µZ ¶ 1 µZ ¶ 1
2 2 2
2
kuvk L 1 ≤ kuk L 2 kvk L 2 , i.e. |uv| ≤ u v .
Ω Ω Ω
(iii) Minkowski inequality asserts that
ku + vk p ≤ kuk p + kvk p .
L L L
p
p 0
(iv) Riesz Theorem: the dual space of L , denoted by (L ) ,can be identi-
0
p
fied with L (Ω) where 1/p +1/p =1 provided 1 ≤ p< ∞. The result is false
0
if p = ∞ (and hence p =1). The theorem has to be understood as follows: if
0
p 0
ϕ ∈ (L ) with 1 ≤ p< ∞ then there exists a unique u ∈ L p 0 so that
Z
p
hϕ; fi = ϕ (f)= u (x) f (x) dx, ∀f ∈ L (Ω)
Ω
and moreover
kuk p 0 = kϕk p 0 .
L (L )
p
(v) L is separable if 1 ≤ p< ∞ and reflexive (which means that the bidual
p
p
p 00
of L , (L ) ,can be identified with L )if 1 <p < ∞.
(vi) Let 1 ≤ p< ∞. The piecewise constant functions (also called step
functions if Ω ⊂ R), or the C ∞ (Ω) functions (i.e., those functions that are
0
p
p
C ∞ (Ω) and have compact support) are dense in L .More precisely if u ∈ L (Ω)
then there exist u ν ∈ C ∞ (Ω) (or u ν piecewise constants) so that
0
lim ku ν − uk L p =0 .
ν→∞
The result is false if p = ∞.
p
0
p 0
Remark 1.14 We will always make the identification (L ) = L . Summariz-
ing the results on duality we have
p 0
(L ) = L p 0 if 1 <p < ∞,
¡ 2 0 2 ¡ 1 0 1 ∞ 0
¢
¢
L = L , L = L ,L ⊂ (L ) .
∞
6=