Page 246 - DMTH505_MEASURE_THEOREY_AND_FUNCTIONAL_ANALYSIS

Unit 22: The Uniform Boundedness Theorem

T x T as k (By continuity of T ) Notes
i k iy i
T x T as k (By continuity of norm)
i k iy
T y = lt T x
i k i k

n i ( x F )
k n
y F .
n
Thus F contains all its limit point and is therefore closed. Further, if x is any element of B, then
n
by hypothesis (c) of the theorem a real number k 0 s.t.
T x k i
i
Let n be a positive integer s.t. n > k. Then
T x < n i
i
so that x F .
n

Consequently, we have B = F .
n
n 1
Since B is complete, it therefore follows by Baire’s theorem that closure of some F , say F n o F ,
n n o
possesses an interior point x . Thus we can find a closed sphere S with centre x and radius r
o o o o
such that S F .
o n o
Now if y is any vector in T (S ), then
i o
y = T
i s o
where s S F .
o o n o
y = Ts n .
i o o
Thus norm of every vector in T (S ) is less than or equal to n . We write this fact as T (S ) n .
i o o i o o
S o x o
Let S = . Then S is a closed unit sphere centred at the origin in B and
r
o
S x
T (S) = T o o
i i r o
1
= T (S ) T (x )
o
i
i
o
r o
1
T (S ) T (x )
r i o i o
o
2n o i
r
o
2n
Hence T o i
i
r
o
This completes the proof of the theorem.

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