Norm of a Function or Square integrable functions
We know that functions can be treated as infinite dimensional vectors. A vector has a norm or length defined for it. How do we do that for a function?
Square integrable functions
Assume that we have a function: $f: \mathbb{R} \to \mathbb{C}$. It is said to be square integrable if it satisfies the following condition:
$$
\int_{-\infty}^{\infty} |f(x)|^2 dx < \infty
$$
This is similar to the squared sum of all elements in a vector and can be considered equivalent to a squared Euclidean norm ($\sum_i v_i$). This is a measure of length and the condition above is asserting that this “length” is finite for the given function (vector). For functions, this quantity is typically called energy.
The set of such functions are commonly denoted as $\mathbb{L}^2$ and is a subset of the vector space of all functions.