Page 275 - DMTH505_MEASURE_THEOREY_AND_FUNCTIONAL_ANALYSIS

Measure Theory and Functional Analysis

Notes n
Let s n = (x, f ) f i
i
i 1
As shown for the case above for (s ), let
n
s s in H where we can take
n

s = (x, f ) f n .
n
n 1
We prove that s = s . Given > 0 we can find n such that n n .
0 0
2
x, e < , s – s < , s s < … (1)
2
i n
n
i n o 1
For some positive integer m > n , we can find all the terms of s in s also.
0 0 n m o
Hence s s contains only finite number of terms of the type (x, e ) e for i = n + 1, n + 2, …
m o n o i i 0 0
2 2
Thus, we get s s x, e so that we have
m o n o i
i n o 1
s s < … (2)
m o n o

Now s – s = s s m o s m o s n o s n o s

s s m 0 s m 0 s n 0 s n 0 s

< + + = 3 (Using (1) and (2))

Since > 0 is arbitrary, s – s = 0 or s = s .
Now consider
(x – (x, e ) e , e) = (x – s, e)
i i j j
But (x – s, e) = (x, e) – (s, e)
j j j
= (x, e) – (lim s , e) … (3)
j n j
By continuity of inner product, we get
(lim s , e) = lim (s , e) … (4)
n j n j
Using (3) in (4), we obtain

(x – (x, e ) e , e) = (x, e) – lim (s , e)
i i j j n j
If e S, then
j
n
(s , e) = (x, e ) e , e = 0
n j i i j
i 1
lim (s , e ) = 0
j
n
n

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