Page 277 - DMTH505_MEASURE_THEOREY_AND_FUNCTIONAL_ANALYSIS

Measure Theory and Functional Analysis

Notes Theorem 6: A orthonormal set in a Hilbert space is linear independent.
Proof: Let S be an orthonormal set in a Hilbert space H.
To show that S is linearly independent, we have to show that every finite subset of S is linearly
independent.
Let S = {e , e , …, e } be any finite subset of S.
1 1 2 n
Now let us consider
e + e + … + e = 0 … (1)
1 1 2 2 n n
Taking the inner product with e (1 k n),
k
n n
e , e = (e , e ) … (2)
i i k i i k
i 1 i 1
Using the fact that (e , e ) = 0 for i k and (e , e ) = 1, we get
i k k k
n
(e , e ) = … (3)
i j k k
i 1
It follows from (2) on using (1) and (3) that
(0, e ) =
k k
= 0 k = 1, 2, …, n.
k
S is linearly independent.
1
This completes the proof of the theorem.

Example: If {e } is an orthonormal set in a Hilbert space H, and if x, y are arbitrary vectors
i
in H, then (x, e )(y, e ) x y .
i i

Solution: Let S e :(x, e )(y, e ) 0
i
i
i
Then S is either empty or countable.
If S is empty, then we have
(x, e )(y, e ) = 0 i
i i
and in this case we define

(x, e )(y, e ) to be number 0 and we have 0 x y .
2
2
i
i
If S is non-empty, then S is finite or it is countably infinite. If S is finite, then we can write
S = {e , e , …, e } for some positive integer n.
1 2 n
In this case we define
n
(x, e )(y, e ) = (x, e )(y, e )
i i i i
i 1

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