Page 184 - DMTH505_MEASURE_THEOREY_AND_FUNCTIONAL_ANALYSIS

Unit 15: Banach Space: Definition and Some Examples

Notes
Since x, y M , there exist sequences (x ) and (y ) in M such that
n n
x x and y y,
n n
By joint continuity of addition and scalar multiplication in M.
x + y x + y for every scalars and .
n n
Since x + y M, we conclude that
n n
x + y M and consequently M is a subspace of N.
This completes the proof of the theorem.

Notes

1. The scalars , can be assumed to be non-zero.
For if = 0 = , then
x + y = 0 M M

2. In a normed linear space, the smallest closed subspace containing a given set of
vectors S is just the closure of the subspace spanned by the set S. To see this, let S be
the subset of a normed linear space N and let M be the smallest closed subspace of N,
containing S. We show that M = [S], where [S] is the subspace spanned by S.

By theorem, [S] is a closed subspace of N and it contains S.
Since M is the smallest closed subspace containing S, we have

M [S].

But [S] M and M = M , we must have

[S] M = M so that [S] M.

Hence [S] = M.

15.1.4 Complete Normed Linear Space

Definition: A normed linear space N is said to be complete if every Cauchy sequence in N
converges to an element of N. This means that if x – x 0 as m, n , then there exists x
m n
N such that
x – x 0 as n .
n
15.1.5 Banach Space

Definition: A complete normed linear space is called a Banach space.
OR
A normed linear space which is complete as a metric space is called a Banach space.

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