Page 58 - DMTH505_MEASURE_THEOREY_AND_FUNCTIONAL_ANALYSIS

Unit 5: Spaces, Hölder

Notes

Note The symbol L is used for such classes when limits of integration are known and
p
mentioning of interval is not necessary.
5.1.2 Conjugate Numbers

1 1
Let p, q be any two n on-negative extended real numbers s.t. 1 , then p, q are called
p q
(mutually) conjugate numbers.
Obviously, 2 is self-conjugate number.
Also if p 2, then q 2. Further, if p = , then q = 1 1, are conjugate numbers.

Note Non-negativity p 1, q 1.

P
5.1.3 Norm of an Element of L -space
The p-norm of any f L [a, b], denoted by f , is defined as
p
p
1
b p
f = |f| p , 0 < p < .
p
a
p
p
Theorem 1: If f L [a, b] and g f, then g L [a, b].
Proof: Let be any positive real number.
{x [a, b] : g (x) > } = {x [a, b] : < g (x) f (x)} ( g f)

= {x [a, b] : f (x) > }
Again f L [a, b]
p
f is measurable over [a, b].

{x [a, b] : f (x) > } is a measurable set.
{x [a, b] : g (x) > } is a measurable set.
g is a measurable function over [a, b]
Again since g (x) f (x), x [a, b]

b b
p
p
|g| dx |f| dx ( |f| L [a, b])
p
a a
b
p
or |g| dx
a
Thus |g| L [a, b].
p

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