quasinorm

A quasinorm is a generalization of a norm used in functional analysis to define quasi-normed or quasi-Banach spaces. Like a norm, a quasinorm assigns a nonnegative size to each vector and is homogeneous, but it relaxes the triangle inequality.

Formally, let X be a vector space over the reals or complexes and let ||·||: X → [0, ∞) be a function. It is called a quasinorm if (i) ||x|| = 0 if and only if x = 0, (ii) ||α x|| = |α| ||x|| for all scalars α, and (iii) there exists a constant K ≥ 1 such that for all x, y in X, ||x + y|| ≤ K (||x|| + ||y||). When K = 1 the quasinorm is a norm.

A standard example arises from L^p spaces for 0 < p ≤ 1, where ||f||_p = (∫ |f|^p)^{1/p} (or the corresponding sequence norm) behaves as a quasinorm rather than a norm. In this range one has the subadditivity ||f + g||_p^p ≤ ||f||_p^p + ||g||_p^p, which yields a relaxed triangle inequality. In contrast, for p ≥ 1 the same function defines a norm.

Quasinorms underpin quasi-Banach spaces, which may lack local convexity and thus differ from Banach spaces in several functional-analytic properties. They appear in harmonic analysis, interpolation, sparse representations, and other areas of analysis where nonconvex behavior is relevant.