Lp norm

A norm answers the question ‘how big is this object?’ with a single non-negative number. For functions there are many reasonable answers, and the L^p norm is a one-parameter dial among them. Turning the dial to p = 1 measures total accumulated size (area under |f|); to p = 2 it measures a root-mean-square or ‘energy’ size; as p grows it pays ever more attention to the tallest spikes and ever less to broad low regions.

For a measurable function f on (X, mu) and 1 <= p < infinity, the L^p norm is ||f||_p = (the integral of |f|^p) raised to the power 1/p. The outer 1/p exponent is essential: it makes the norm homogeneous, so ||c f||_p = |c| ||f||_p for any scalar c, restoring the scaling a norm must have. Without it, scaling f by c would multiply the quantity by |c|^p instead of |c|.

Strictly this is only a norm after identifying functions equal almost everywhere, because ||f||_p = 0 forces f = 0 almost everywhere, not everywhere. On the raw space of functions it is merely a seminorm. The triangle inequality ||f + g||_p <= ||f||_p + ||g||_p holds for all p >= 1 and is exactly Minkowski's inequality; for 0 < p < 1 it fails, so those exponents do not give a genuine norm.