quasinorms

Quasinorms are a generalization of norms used in functional analysis. A quasinorm on a vector space X over the real or complex numbers is a function ||·||: X → [0, ∞) that satisfies three conditions: ||x|| ≥ 0 and ||x|| = 0 if and only if x = 0; homogeneity ||αx|| = |α| ||x|| for all scalars α; and a relaxed triangle inequality: there exists a constant C ≥ 1 such that ||x + y|| ≤ C (||x|| + ||y||) for all x, y ∈ X. If the constant C can be taken as 1, the function is a norm; thus every norm is a quasinorm, but not conversely.

Quasinorms generalize norms by allowing the triangle inequality to be violated in a controlled way. They are central to the study of quasi-Banach spaces, which are complete spaces equipped with a quasinorm. Unlike normed spaces, quasi-Banach spaces need not be locally convex, and their duality theory can be more delicate. The unit ball of a quasinormed space need not be convex.

Examples and typical settings include the L^p spaces for 0 < p ≤ 1, where the quasi-norm ||f||_p = (∫ |f|^p)^{1/p} is finite for measurable functions. For p < 1, this functional fails the usual triangle inequality but satisfies a quasi-triangle inequality, making L^p into a quasi-Banach space. Similar quasi-norms arise in sequence spaces and various function spaces used in harmonic analysis and interpolation theory.

Notes: Quasinorms do not arise from inner products in general; consequently, many classical results relying on convexity and the parallelogram identity do not apply. They provide a flexible framework for analysis beyond normed spaces.