Inner product of functions

1. Apr 25, 2005

sid_galt

What is the proof that the inner product of two functions f(x) and g(x) is

$$\int_{a}^{b} f(x)g(x)dx$$

Or is this actually a definition of the inner product for functions? If it is a definition, then what is it based on?

Thank you

Last edited: Apr 25, 2005
2. Apr 25, 2005

arildno

You've gotten it backwards:
That particular integral can be shown to fulfill the PROPERTIES OF AN INNER PRODUCT.

For your info, if w(x) is a positive integrable function, the following can also be shown to be an inner product:
$$<f,g>\mid_{w}=\int_{a}^{b}f(x)g(x)w(x)dx$$

3. Apr 25, 2005

Lonewolf

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