Page 86 - DMTH505_MEASURE_THEOREY_AND_FUNCTIONAL_ANALYSIS

Unit 7: Convergence and Completeness

Notes
1
p p
lim f n f dt = 0
n
lim f n f = 0.
n p
Theorem: In a normed linear space, every convergent sequence is a Cauchy sequence.
Proof: Let the sequence <x > in a normed linear space N, converges to a point x N . We shall
n o 1
show that it is a Cauchy sequence.
Let > 0 be given. Since the sequence converges to x a positive integer m s.t.
o o
n m x – x < /2 … (1)
o n o
Hence for all m, n m , we have
o
x – x = x – x + x – x
m n m o o n
x – x + x – x
m o o n
< = by (1)
2 2

It follows that the convergent sequence <x > is a Cauchy sequence.
n
Theorem: Prove that L [0, 1] is complete.
Proof: Let (f ) be any Cauchy sequence in L , and let
n
A = {x : |f (x)| > f },
k k k
B = {x : |f (x) – f (x)| > f – f }.
m, n k m k m
Then m (A ) = 0 = m (B ) (k, m, n = 1, 2, 3, …),
k m, n
So that if E is the union of these sets, we have m (E) = 0.

Now, if x F = [0, 1] – E, then
|f (x) f
k k
|f (x) – f (x) f – f 0 as n, m .
n m n m
Hence the sequence (f ) converges uniformly to a bounded function on F.
n
Define f : [0, 1] R by

lim f (x) if x F
n
f (x) = n
0, if x E
Then f L and f – f 0 as n .
n
Thus L is

Hence proved.

7.2 Summary

 A sequence <x > in a normal linear space X with norm . is said to converge to an element
n
x X if for arbitrary > 0, however small, n N s.t. x – x < , n > n . Then we write
o n o
lim x n x .
n

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