Page 188 - DMTH505_MEASURE_THEOREY_AND_FUNCTIONAL_ANALYSIS

Unit 15: Banach Space: Definition and Some Examples

2
x – z < 2 Notes
m
x – z <
m
n
n
It follows that the Cauchy sequence <x > converges to z C (or R ).
m
Hence C or R are complete spaces and consequently they are Banach spaces.
n
n
15.2 Summary
 A linear space N together with a norm defined on it, i.e. the pair (N, ) is called a normed
linear space.

 Let (N, ) be a normed linear space. A sequence (x ) in N is said to converge to an element
n
x in N if given > 0, there exists a positive integer n such that
o
x – x < for all n n .
n o
 If N is a normed linear space, then
x y x y for any x, y N.

 A normed linear space N is said to be complete if every Cauchy sequence in N converges
to an element of N.
 A complete normed linear space is called a Banach space.

15.3 Keywords

A Subspace M of a Normed Linear Space: A subspace M of a normed linear space is a subspace of
N consider as a vector space with the norm obtain by restricting the norm of N to the subset M.
If norm on M is said to be induced by the norm on N. If M is closed in N, then M is called a closed
subspace of N.
Banach Space: A complete normed linear space is called a Banach space.
Complete Normed Linear Space: 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
m n
there exists x N such that
x – x 0 as n .
n
Normed Linear: A linear space N together with a norm defined on it, i.e., the pair (N, ) is called
a normed linear space and will simply be denoted by N for convenience.

15.4 Review Questions

1. Let N be a non-zero normed linear space, prove that N is a Banach space {x : x = 1} is
complete.
2. Let a Banach space B be the direct sum of the linear subspaces M and N, so that B = M N.
If z = x + y is the unique expression of a vector z in B as the sum of vectors x and y in M and
N, then a new norm can be defined on the linear space B by z = x + y .
Prove that this actually is a norm. If B symbolizes the linear space B equipped with this
new norm, prove that B is a Banach space of M and N are closed in B.

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