What is the basis for the inner product of functions?

[sid_galt]

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

<br /> \int_{a}^{b} f(x)g(x)dx<br />

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

 

[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:
&lt;f,g&gt;\mid_{w}=\int_{a}^{b}f(x)g(x)w(x)dx

 

[Lonewolf]

As for its basis, http://mathworld.wolfram.com/InnerProduct.html will explain this.