Minkowski's inequality

Minkowski's inequality is simply the triangle inequality for the L^p norm: the size of a sum of two functions never exceeds the sum of their sizes. Geometrically it says that, just as you cannot shorten a journey by taking two sides of a triangle instead of one, you cannot make ||f + g||_p larger than ||f||_p + ||g||_p. Without this fact the L^p norm would not deserve the name ‘norm’ at all.

Formally, for 1 <= p <= infinity and f, g in L^p, one has ||f + g||_p <= ||f||_p + ||g||_p. The standard proof for 1 < p < infinity writes the integral of |f+g|^p as the integral of |f+g| times |f+g|^(p-1), splits |f+g| <= |f| + |g|, and applies Hölder's inequality to each piece with the conjugate exponent q = p/(p-1). The endpoint cases p = 1 and p = infinity follow directly from the ordinary triangle inequality for numbers.

Equality holds (for 1 < p < infinity) only when f and g point in the ‘same direction’ — precisely, when one is a non-negative scalar multiple of the other almost everywhere. A caveat: the inequality genuinely requires p >= 1. For 0 < p < 1 it reverses on suitable functions, which is why those exponents fail to produce a normed space.