Inner product space axiom

Question

So, I just have a small question.

Given that $\langle f,g\rangle=\langle g,f\rangle$, $\forall f, g\in V$. One of the Inner Product Space axioms.

So, I prove by saying $\langle f,g\rangle = \int f(x)g(x)dx$ and $\langle g,f\rangle = \int g(x)f(x)dx$. So these are obviously equivalent, but how do I justify that the order of $g(x)$ and $f(x)$ in the integral doesn't matter?

Thanks in advance :)

Answer 1

The commutative law of multiplication.

Answer 2

To extend ncmathsadist's answer, the multiplication inside the integral is pointwise multiplication. That is, we calculate $f(x)$ and $g(x)$ and multiply these numbers as real numbers.

The multiplication in the real numbers is commutative, so the pointwise multiplication of real-valued functions is commutative.