Page 183 - DMTH505_MEASURE_THEOREY_AND_FUNCTIONAL_ANALYSIS

Measure Theory and Functional Analysis

Notes 15.1.2 Convergent Sequence in Normed Linear Space

Definition: Let (N, ) be a normed linear space. A sequence (x ) is N is said to converge to an
n
element x in N if given > 0, there exists a positive integer n such that
o
x – x < for all n n .
n o
If x converges to x, we write Lim x x .
n n n
or x x as n
n
It follows from the definition that

x x x – x 0 as n
n n
Theorem 2: If N is a normed linear space, then

x y x – y for any x, y N
Proof: We have

x = (x – y) + y
x – y + y
x – y x – y … (1)
Using (1), we have

– ( x – y ) = y – x y – x
But y – x = (–1) (x – y) = |–1| x – y
Therefore
– ( x – y ) x – y so that

x – y – x – y … (2)
From (1) and (2) we get

x y x – y
This completes the proof of the theorem.

15.1.3 Subspace of a Normed Linear Space

Definition: 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. This 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.
Theorem 3: Let N be a normed linear space and M is a subspace of N. Then the closure M of M is
also a subspace of N.

(Note that since M is closed, M is a closed subspace).

Proof: To prove that M is a subspace of N, we must show that any linear combination of
element in M is again in M. That is if x and y M , then x + y M for any scalars and .

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