Page 70 - DMTH505_MEASURE_THEOREY_AND_FUNCTIONAL_ANALYSIS

Unit 5: Spaces, Hölder

5.2 Summary Notes

p
 The class of all measurable functions f (x) is known as L – space over [a, b], if Lebesgue-
integrable over [a, b] for each p exists, 0 < p < , i.e.
b
p
|f| dx , (p 0)
a
p
 The p-norm of any f L [a, b], denoted by f , is defined as
p
1
b p
f |f| p , 0 p
p
a

1 1
 Let p, q > 1 be such that 1 , and let u and v be two non-negative numbers, at least
p q
one being non-zero. Then the function f : [0, 1] R defined by

1
f(t) ut v 1 t q q , t [0, 1],

has a unique maximum point at
1
u p q
s
u p v p
q
p
 Let p and q be conjugate indices or exponents and f L [a, b], g L [a, b], then it is evident
that
(i) f, g L [a, b]
(ii) fg f g i.e.
p q
1 1
|fg| |f| p p |g| q q

5.3 Keywords

1 1
Conjugate Numbers: Let p, q be any two n on-negative extended real numbers s.t. 1 , then
p q
p, q are called (mutually) conjugate numbers.
Hölder's Inequality: Let a , a , …, a , b , b , …, b be non-negative numbers. Let p, q > 1 be real
1 2 n 1 2 n
1 1
number with the property 1 . Then
p q

1 1
n
n p n q
a b p q
j j a b
j j
j 1
j 1 j 1
p
q
Moreover, one has equality only when the sequences a , , a p and b , , b q are proportional.
1 n n
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