Inner Product of Functions
The inner/dot product for Euclidean vectors is defined as:
$$
\langle u, v \rangle = \sum_{i} u_i v_i
$$
Following on from how we defined a norm for square integrable functions, we can similarly define an inner product for such functions in $\mathbb{L}^2$:
$$
\langle f, g \rangle = \int_{-\infty}^{\infty} f(x) g(x) dx
$$
This inner product can now be used to define a norm/length for our space:
$$
|| f || = \sqrt{ \langle f, f \rangle } = \sqrt{\int_{-\infty}^{\infty} |f(x)|^2 dx}
$$