Page 84 - DMTH505_MEASURE_THEOREY_AND_FUNCTIONAL_ANALYSIS

Unit 7: Convergence and Completeness

Notes
n n
S – S = f i f
n m p i
i m 1 i m 1
f i
i n o
Sequence <S > of partial sums is a Cauchy sequence
n
<S > converges.
n
Sequence <f > is summable to some element S X.
n
But X is a complete space. Therefore <S > will converge to some element S X.
n
Sufficient part: Given that every absolutely summable sequence in the space X is summable.

To show that X is complete.
Let <f > be a Cauchy sequence in X.
n
For each positive integer k, we can choose a number n N such that
k
1
f – f < , n, m n … (ii)
n m k k
2
We can choose these n ’s such that n > n .
k k+1 k
Then f is a subsequence of <f >.
n n
k 1
Setting g = f and g = f f , (k > 1), we get a sequence <g > s.t. its k partial.
th
1 n 1 k n k n k 1 k
Sum = S = g + g + … + g = f (f f ) (f f ) f .
k 1 2 k n 1 n 2 n 1 n k n k 1 n k
1
Now, g = f f , [by (ii)], k > 1
k n k n k 1 k 1
2
1
g k g 1 k 1 = g + 1 (a finite quantity)
1
k 1 k 2 2
The sequence <g > is absolutely summable and hence by the hypothesis, it is a summable
k
sequence.
The sequence of partial sums of this sequence converges to some S X.

The sequence <S > converges and hence f converges to some f X.
k n k
Now, we shall show that the limit f = f.
n
Again, since <f > is a Cauchy sequence, we get that for each > 0, however small, n N s.t.
n
n, m > n .

f – f < .
n m 2

Also since f f, n N such that k n ,
n k

LOVELY PROFESSIONAL UNIVERSITY 77