Page 80 - DMTH505_MEASURE_THEOREY_AND_FUNCTIONAL_ANALYSIS

Unit 7: Convergence and Completeness

7.1.2 Cauchy Sequence Notes

Definition: A sequence <x > in a normal linear space (X, . ) is said to be a Cauchy sequence if for
n
arbitrary > 0, n N s.t.
0
x – x < , n, m n .
n m 0
7.1.3 Complete Normed Linear Space

Definition: A normed linear space (X, . ) is said to be complete if every Cauchy sequence <x > in
n
it converges to an element x X.
7.1.4 Banach Space

Definition: A complete normed linear space is also called Banach space.

7.1.5 Summable Series

Definition: A series u n in N is said to be summable to a sum u if u N and lim S u , where
1
1
n n
n 1
S = u + u + … + u
n 1 2 n
In this case, we write

u u .
n
n 1

Further, the series u is said to be absolutely summable if u n .
n
n 1 n 1

7.1.6 Riesz-Fischer Theorem

p
Theorem: The normed L -spaces are complete for (p 1).
Proof: In order to prove the theorem, we shall show that every Cauchy sequence in L [a, b] space
p
p
p
converges to some element f in L -space. Let <f > be one of such sequences in L -space. Then for
n
given > 0, a natural number n , such that
0
m, n n f – f < ,
0 m n p
1
since is arbitrary therefore taking , we can find a natural number n such that
2 1
1
for all m, n n f – f <
1 m n p
2
1
Similarly, taking , k N, we can find a natural number n , such that
2 k k
1
for all m, n n f – f <
k m n p 2 k

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