What are the boundaries in inner product of functions

Question

As we know the definition of an inner product is: $$ \int_a^bf(x)^*g(x) dx $$ int quantum physics we chose $\infty$ for a & b (negative and positive). But for a general mathematical purpose how do we choose a and b? since choose differently would cause different answer for the inner poroduct?

Answer 1

It is more than a "different inner product" when you change the values of $a$ and $b$.

If $a=-\infty$ and $b=\infty$, the integral defines an inner product on the space of all squared integrable functions on the real line: $$ H = \left\{f:\mathbb{R}\to\mathbb{C}\mid \int_{\mathbb{R}} |f|^2<\infty\right\} $$

If $a<b$ are two real numbers, then the underlying vector space becomes: $$ H = \left\{f:[a,b]\to\mathbb{C}\mid \int_a^b |f|^2<\infty\right\} $$

Choosing the values of $a$ and $b$ depends on the domain of the functions that you want to work with. For instance, if you want to consider periodic functions on $[0,2\pi]$, then $a=0$ and $b=2\pi$.

For fixed values of $a$ and $b$, as GEdgar mentioned in the comment, there are weighted versions of inner products.