Page 83 - DMTH505_MEASURE_THEOREY_AND_FUNCTIONAL_ANALYSIS

Measure Theory and Functional Analysis

Notes
1
b p
p
lim g f dx = 0
m m
a
lim g f = 0
m m p
lim f f = 0 as g = f
m n m p m n m

f f < .
n m
p
Also f f < .
m n m
p
f – f = f f f f
m p m n m n m p
f m f n m p f n m f p

< ( + ) = .
Hence lim f f = 0
m m p
or lim f = f L [a, b].
p
m m
This proves the theorem.

Alternative Statement of this Theorem

p
p
A convergent sequence <f > in L -spaces has a limit in L -space.
n
Or
p
p
Every Cauchy sequence <f > in the L -space converges to a function in L -space.
n
Theorem: Prove that a normed linear space is complete iff every absolutely summable sequence
is summable.
Proof: Necessary part
Let X be a complete normed linear space with norm . and <f > be an absolutely summable
n
sequence of elements of X

f = M < ,
n
n 1
For arbitrary > 0, however small, n N
o

s.t. f < , … (i)
n
n n o
n
Now, if S = f i , then n m n , we get
n o
i 1

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