Definition of the inner product of functions I am taking a linear algebra course and we have come across the inner product of two functions. The textbook by Anton defines it as:
$\int_{a}^{b} f(x)g(x) dx$
I understand this is handy, as it obeys the necessary axioms. I have also looked at this question. However, I am still confused why the textbook called this inner product the inner product. Is this the only inner product for functions?

It is not the only inner product on functions. For example, $$\langle f,g\rangle=\int_a^bf(x)g(x)e^{-x^2}\,dx$$ is also an inner product. There are many, many ways we can construct inner products of functions. Probably your textbook means that this is the inner product the author(s) will use throughout the book.
