Page 187 - DMTH505_MEASURE_THEOREY_AND_FUNCTIONAL_ANALYSIS

Measure Theory and Functional Analysis

Notes = x + y x + x + y y
= ( x + y ) ( x + y ).
If x + y = 0, then the above inequality is evidently true.
If x + y 0, we can divide both sides by it to obtain

x + y x + y .

1 1
n 2 n 2
(iii) x = | x | 2 | | |x | 2
i i
i 1 i 1
= | | x .
This proves that R or C are normed linear spaces.
n
n
n
Now we show the completeness of C (or R ).
n
n
Let < x , x , … x > be a Cauchy sequence in C (or R ). Since each x is an n-tuple of complex (or
n
1 2 n m
real) numbers, we shall write
x = x (m) , x (m) , , x (m)
m 1 2 n
th
So that x (m) is the k coordinate of x .
k m
Let > 0 be given, since <x > is a Cauchy sequence, there exists a positive integer m , such that
m o
 , m m x x
o m 
2
x x 2
m 
n
x (m) x ( )  2 … (1)
i i
i 1

x (m) x ( )  2 (i = 1, 2, ……, n)
i
i
x (m) x ( ) 
i i

Hence x (m) m 1 is a Cauchy sequence of complex (or real) numbers for each fixed but
i
arbitrary i.
Since C (or R) is complete, each of these sequences converges to a point, say 2 in C (or R) so that
i
Lim x (m) = z (i = 1, 2, …, n) … (2)
m i i
n
Now we show that the Cauchy sequence <x > converges to the point z = (z , z , ……, z ) C (or
m 1 2 n
R ).
n
To prove this let  in (1). Then by (2) we have
n
2
x (m) z 2
i i
i 1

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