Page 257 - DMTH505_MEASURE_THEOREY_AND_FUNCTIONAL_ANALYSIS

Measure Theory and Functional Analysis

Notes Theorem 5: In a Hilbert space the inner product is jointly continuous i.e.,
x x, y y (x , y ) (x, y)
n n n n
Proof: We have
|(x , y ) – (x, y)| = |(x , y ) – (x , y) + (x , y) – (x,, y)|
n n n n n n
= |(x , y – y) + (x – x, y)|
n n n
(by linearity property of inner product)
|(x , y – y)| + |(x – x, y)| [ | + | | | + | |]
n n n
x + y – y + x – x y [By Schwarz inequality]
n n n
Since x x and y y as n .
n n
Therefore y – y 0 and x – x 0 as h . Also (x ) is a continues sequence, it is bounded
n n n n
so that x M n.
n
Therefore
| (x , y ) – (x, y) | 0 as n .
n n
Hence (x , y ) (x, y) as n .
n n
This completes the proof of the theorem.
Theorem 6: A closed convex set E in a Hilbert space H continuous a unique vector of smallest
norm.

Proof: Let = inf { e ; e E}
To prove the theorem it suffices to show that there exists a unique x E s.t. x = .
Definition of yields us a sequence (x ) in E such that
n
Lim x n = … (1)
n
x x
Convexity of E implies that m n E . Consequently
2

x x
m n x + x 2 … (2)
2 m n
Using parallelogram law, we get
2
x + x + x – x 2 = 2 x + 2 x 2
2
m n m n n n
2
or x – x 2 = 2 x + 2 x – x – x 2
2
m n m n m n
2
2
2 x + 2 x – d 2 (Using (2))
m n
0 as m, n (Using (1))
x – x 2 0 as m, n
m n
(x ) is a CAUCHY sequence in E.
n
x E such that Lim x x , since H is complete and E is a closed subset of H, therefore
n n
E is also complete and consequently (x ) is in E is a convergent sequence in E.
n

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