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Mathematics-Online course: Linear Algebra - Basic Structures - Scalar Product and Norm

Properties of Scalar Products

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For the vector space of continuous real-valued functions on $ [0,1]$ a scalar product can be defined by

$\displaystyle \langle f,g\rangle := \int\limits_0^1 f(x)g(x)\,dx\,. $

S1 holds true, since a continuous function not vanishing at some point does not vanish in a (small enough) neighbourhood of this point. Thus, the integral $ \int f^2$ cannot equal zero.

Property S2 results from the definition of multiplication of functions.

Properties S3 and S4 are satisfied, since integration is linear, that is, integration has the respective properties.

Introducing a positive weight function $ w$ we can generalise the definition of the scalar product:

$\displaystyle \langle f,g\rangle_w := \int\limits_0^1 fg\,w\,. $

For example, the weighted scalar products

$\displaystyle \int_0^1 f(r)g(r) r\,dr,\quad \int_0^1 f(r)g(r) r^2\,dr $

are of importance in connection with radial-symmetric functions on a disk or in a sphere.
(Authors: Burkhardt/Höllig/Hörner)
  automatically generated 4/21/2005