Page 253 - DMTH505_MEASURE_THEOREY_AND_FUNCTIONAL_ANALYSIS

Measure Theory and Functional Analysis

Notes

Notes
1. Part (b) shows an inner product is conjugate linear in the second variable.

2. If (x, y) = 0 x X, then y = 0. If (x, y) = 0 x X, it should be true for x = y also,
so that (y, y) = 0 y = 0.

n
Example 1: The space  is an inner product space.
2
n
Solution: Let x = (x , x , ……, x ), y = (y , y , ……, y )  .
1 2 n 1 2 n 2
n
Define the inner product on  as follows:
2
n
(x, y) = x y i
i
i 1
Now
n
(i) ( x + y, z) = x i y i i z
i 1
n n
= x i i z y i i z
i 1 i 1
= (x, z) + (y, z)

n
(ii) (x,y) = x y i
i
i 1
= (x y x y  x y )
1 1 2 2 n n
= (x y 1 2 x y 2  x y )
n
1
n
= x 1y 1 2 x y 2  x y n
n
= (y, x)
n
(iii) (x, x) = x i x
i
i 1
n
2
= x i 0
i 1
Hence (x, x) 0 and (x, x) = 0 x = 0 for each i, i.e. (x, x) = 0 x = 0.
i
n
(i) – (iii)  is a inner product space.
2
23.1.2 Hilbert Space and its Basic Properties

By using the inner product, on a linear space X we can define a norm on X, i.e. for each x X, we
define x = (x,x) . To prove it we require the following fundamental relation known as
Schwarz inequality.

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