What is the Inner Product Space for Square-Integrable Functions?

[mekarim]

http://en.wikipedia.org/wiki/Square-integrable_function


According to the tutorial: it says
g*(x) is the complex conjugate of g

but I can't get the idea from where this g(x) function comes, than why is it the complex conjugate?

And it seems i can't visualize the inner product space? Some practical example would help me a lot.

Thanks!

 

[olivermsun]

The idea of square integrable functions is that the integral of the squared magnitude converges. For complex valued functions, |f(x)|^2 = ∫ f(x) f*(x) dx, which suggests a natural way to define both the "norm" and the "product" in the space of square integrable functions. You just say that the inner product <f, g> has to satisfy the property that |f|^2 = <f, f> and therefore <f, g> = ∫ f(x) g*(x) dx.

 

[HallsofIvy]

In particular, you want $|f|= <f , f>$. Since the "norm" is defined as $\int f(x)f^*(x)dx= <f,f>$ the natural way to define the "inner product" of two such functions, f and g, is $<f, g>= \int f(x)g^*(x)dx$.