Page 185 - DMTH505_MEASURE_THEOREY_AND_FUNCTIONAL_ANALYSIS

Measure Theory and Functional Analysis

Notes In the definition of a Banach space completeness means that if
x – x 0 as m, n , where (x ) N, then
m n n
a x N such that
x – x 0 as n .
n

Note A subspace M of a Banach space B is a subspace of B considered as a normed linear
space. We do not require M to be complete.

Theorem 4: Every complete subspace M of a normed linear space N is closed.
Proof: Let x N be any limit point of M.
We have to show that x M.
Since x is a limit point of M, there exists a sequence (x ) in M and x x as n .
n n
But, since (x ) is a convergent sequence in M, it is Cauchy sequence in M.
n
Further M is complete (x ) converges to a point of M so that x M.
n
Hence M is closed.
This completes the proof of the theorem.

Theorem 5: A subspace M of a Banach space B is complete iff the set M is closed in B.
Proof: Let M be a complete subspace of a Banach space M. They be above theorem, M is closed
(prove it).

Conversely, let M be a closed subspace of Banach space B. We shall show that M is complete.
Let x = (x ) be a Cauchy sequence in M. Then
n
x x in B as B is complete.
n
We show that x M.

Now x M x M ( M being closed M = M )
Thus every Cauchy sequence in M converges to an element of M. Hence the closed sequence M
of B is complete. This completes the proof of the theorem.

Example 1: The linear space R of real numbers or C of complex numbers are Banach
spaces under the norm defined by
x = |x|, x R (or C)
Solution: We have

x = |x| > 0 and x = 0 |x| = x = 0
Further, let z , z C and let z and z be their complex conjugates, then
2
1
1 2
2
|z + z | = (z + z ) (z 1 2 z )
1 2 1 2
= z z 1 z 1 2 z z 2 1 z z 2 2 z
1
2 2
z 2 z 2 z z
1 1 2
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