I have a quite simple question, which I'm not really able to answer. Assume that you have to functions $f,g$ on an infinite dimensional function (normed) space. Define the usual inner product on a given interval $[0,b]$ as $$\langle f,g \rangle = \int_{0}^{b} f(t)g(t)dt$$
Now assume that I would like to approximate the function $f$ by a suitable finite subset of basis functions $\{\phi_i\}_{i}^{N}$, which can be done by using the aforementioned inner product.
Ok, my question is the following: Can I define this inner product if $b\to\infty$? Do I need any hypothesis on the set of $\phi$'s or the function $f$?
Thanks!!!!!
Defining an the inner product on a space of functions on a measure space $(\Omega,\mathcal B_\Omega,dx)$ boils down to Cauchy-Schwarz inequality, which provides the estimate $$ \left|\int_{\Omega}fgdx\right|\le \left(\int_\Omega f^2dx\right)^{\frac{1}{2}}\left(\int_\Omega g^2dx\right)^{\frac{1}{2}}. $$ Hence you could define your inner product on the vector space of continuous square integrable functions on $[0,\infty)$, i.e. $f\in C[0,\infty)$ such that $\int_0^\infty f^2dx<\infty$ (where Cauchy-Schwarz proves linearity of the space). Then to obtain a Hilbert space you need the norm induced by the inner product $$ \|f\|_{L^2}:=\sqrt{\langle f,f\rangle}. $$ to be complete. This can always be done.
I do not see any easy way to use any set of approximating functions on $[0,b]$.