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Inner Product Spaces

An inner product space is a vector space, say , together with a complex bilinear function having the following properties:


(i)

where

(ii)

where
and
are complex numbers
(iii)

if

and

.


(a) The condition is quite necessary, otherwise there would be conflict with (iii). Indeed, if , then

In other words, condition (i) guarantees that the positive definiteness condition (iii) is preserved.

(b) With the help of (i), condition (ii) is equivalent to

 (11)

Thus we see that a complex scalar (say, or ) in the first factor of the inner product gets complex conjugated when it gets separated from the inner product as a multiplicative factor. One says that is linear in the second argument and antilinear in the first argument.

(c) The square root of , , is called the norm of the vector . It is always understood that the norm is finite. In particular

(d) The inner product satisfies the Cauchy-Schwarz inequality

This inequality has a nice geometrical interpretation for real inner product spaces. In that case is the familiar inner product and

The Cauchy-Schwarz inequality follows from the fact that for any complex

Letting we obtain for all real

Consequently, the discriminant,

of this quadratic expression must be negative or zero, otherwise this expression would be negative for some values of . It follows that

(e) The inner product implies the triangle inequality

 (12)

This inequality readily follows from the properties of the inner product (Why?)

Next: Normed Linear Spaces Up: Infinite Dimensional Vector Spaces Previous: Infinite Dimensional Vector Spaces   Contents   Index
Ulrich Gerlach 2010-12-09