Page 87 - DMTH505_MEASURE_THEOREY_AND_FUNCTIONAL_ANALYSIS

Measure Theory and Functional Analysis

Notes  A complete normed linear space is also called Banach space.
p
 The normed L -spaces are complete for (p 1).
p
p
 A convergent sequence <f > in L -spaces has a limit in L -space.
n
 A normed linear space is complete iff every absolutely summable sequence is summable.
 In a normed linear space, every convergent sequence is a Cauchy sequence.

7.3 Keywords

Banach Space: A complete normed linear space is also called Banach space.
Cauchy Sequence: A sequence <x > in a normal linear space (X, . ) is said to be a Cauchy
n
sequence if for arbitrary > 0, n N s.t.
0
x – x < , n, m n .
n m 0
Complete Normed Linear Space: A normed linear space (X, . ) is said to be complete if every
Cauchy sequence <x > in it converges to an element x X.
n
Convergence almost Everywhere: Let <f > be a sequence of measurable functions defined over a
n
measurable set E. Then <f > is said to converge almost everywhere in E if there exists a subset E
n o
of E s.t.
(i) f (x) f (x), x E – E .
n o
and (ii) m (E ) = 0.
o
Convergent Sequence: A sequence <x > in a normal linear space X with norm . is said to
n
converge to an element x X if for arbitrary > 0, however small, n N such that x – x <
0 n
, n > n .
0
Then we write lim x x .
n n
Normed Linear Space: A linear space N together with a norm defined on it, i.e., the pair (N, )
is called a normed linear space.

Summable Series: A series u n in N is said to be summable to a sum u if u N and lim S u ,
1
1
n n
n 1
where
S = u + u + … + u
n 1 2 n
7.4 Review Questions

n
1. Prove that  is complete.
p
2. Prove that the vector space L equipped with is a complete vector space.
L
3. Suppose f L is supported on a set of finite measure.
Then f L for all p < , and
p
f f as p .
L L

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