in class we learned about the inner product of two functions as $\int_{a}^{b} f(x)g(x) dx$ but I failed to get an intuition about what it means, can someone please explain it to me?
also what the norm $\|f(x)\|_2=\sqrt{\int_a^b f^2(x) dx}$ means I can't understand what means that a function has a norm / length / magnitude ? please give me some intuition for these concepts or a way to think about them ?
$\int_a^b |f(x)| dx$ is the area between the axis $y=0$ and the graph of the function $|f|$.
$\int_a^b |f(x)|^2 dx$ is the area between the axis $y=0$ and the graph of $f^2$.
$\int_a^b f(x)g(x) dx$ is the algeabric area between the axis $y = 0$ and the graph of $fg$, 'algeabric' in the sens that area above the axis $y = 0$ is counted positively and area below the axis $y = 0$ is counted negatively.
It turns out that we can use these notions to define a norm and an inner product on spaces of functions.
The norm-2 is particularly important because it is an euclidian norm, meaning that it is induced by an inner product ( $\langle f,g \rangle = \int_a^b f(x)g(x) dx$ ). You can therefore have various notions such as orthogonality for functions.
Does that help?