Page 255 - DMTH505_MEASURE_THEOREY_AND_FUNCTIONAL_ANALYSIS

Measure Theory and Functional Analysis

Notes x + 2 x y + y 2 [using Schwarz inequality]
2
= ( x + y ) 2
Therefore x + y x + y
(iii) x 2 = ( x, x)
= (x, x)
= | | x 2
2
, x = x

(i)-(iii) imply that x = (x, x) is a norm on X. This completes the proof of the theorem.

Note Since we are able to define a norm on X with the help of the inner product, the inner
product space X consequently becomes a normed linear space. Further if the inner product
space X is complete in the above norm, then X is called a Hilbert space.

23.1.3 Hilbert Space: Definition

A complete inner product space is called a Hilbert space.
Let H be a complex Banach space whose norm arises from an inner product which is a complex
function denoted by (x, y) satisfying the following conditions:
H : ( x + y, 2) = (x, 2) + (y, 2),
1
H : (x, y) = (y, x), and
2
2
H : (x, x) = x ,
3
for all x, y, z H and for all , C.

23.1.4 Examples of Hilbert Space

n
1. The space  is a Hilbert space.
2
n
n
We have already shown in earlier example that  is an inner product space. Also  is a
2
2
n
n
Banach space. Consequently  is a Hilbert space. Moreover  , being a finite dimensional,
2
2
n
hence  is a finite dimensional Hilbert space.
2
2.  is a Hilbert space.
2
Consider the Banach space  consisting of all infinite sequence x = (x ), n = 1, 2, … of
2 n
2
complex numbers such that x n with norm of a vector x = (x ) defined by x =
n
n 1
2
x n .
n 1

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