Define the inner product:
$<f,g>=\int^{a}_{b} f(x) g(x) dw(x). \quad (1)$
What does it mean the following statement?
"we assume that $w $ has infinitely many points of increase, in order to ensure that (1) is an inner product"
Could you please explain it? why do we need a weight function?
If $w$ is constant on some subinterval $[c,d] \subseteq [b,a]$, then for any function $f$ supported on $[c,d]$ we have $\langle f, f \rangle = 0$ but $f \not \equiv 0$.