Page 256 - DMTH505_MEASURE_THEOREY_AND_FUNCTIONAL_ANALYSIS

Unit 23: Hilbert Spaces: The Definition and Some Simple Properties

We shall show that if the inner product of two vectors x = (x ) and y = (y ) is defined by Notes
n n
(x, y) = x y , then  is a Hilbert space.
2
n
n
n 1
We first show that inner product is well defined. For this we are to show that for all x, y in  the
2
infinite series x y is convergent and this defines a complex number.
n
n
n 1
By Cauchy inequality, we have
1 1
n n 2 n 2
2 2
x y x y
i i i i
i 1 i 1 i 1
1 1
2 2
2 2
x n y n .
n 1 n 1

n
2 2
Since x n and y n are convergent, the sequence of partial sum x y is a monotonic
i
i
n 1 n 1 i 1
increasing sequence bounded above. Therefore, the series x y is convergent. Hence
i
i
n 1
x y is absolutely convergent having its sum as a complex number.
n
n
n 1
Therefore (x, y) = x y n is convergent so that the inner product is well defined. The condition
n
n 1
of inner product can be easily verified as in earlier example.

Theorem 4: If x and y are any two vectors in a Hilbert space, then
(x + y) + x – y 2 = 2 ( x + y )
2
2
2
Proof: We have for any x and y
(x + y) 2 = (x + y, x + y) (By def. of Hilbert space)
= (x, x + y) + (y, x + y)
= (x, x) + (x, y) + (y, x) + (y, y)
2
= x + (x, y) + (y, x) + y 2 … (1)
x – y 2 = (x – y, x – y)

= (x, x – y) – (y, x – y)
= (x, x) – (x, y) – (y, x) + y 2 … (2)
Adding (1) and (2), we get
2
x + y + (x – y) 2 = 2 x + 2 y = 2 ( x + y )
2
2
2
2
This completes the proof of the theorem.
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