normstos

Normstos are a family of norm-like functionals defined on vector spaces over real or complex numbers, intended to generalize the standard concept of a norm by allowing additional parameterization that can encode directional sensitivity or scale invariance properties. They assign a nonnegative real number to each vector, with core properties including nonnegativity and definiteness (||v||_n = 0 if and only if v = 0), positive homogeneity (||αv||_n = |α| ||v||_n), and a generalized triangle inequality. The triangle inequality for normstos can be stated in various forms depending on the chosen variant; common forms include a subadditivity condition or a relaxed inequality that allows a constant factor, sometimes referred to as a quasi-norm property.

A typical way to construct a normsto is to start with a standard norm and apply a monotone transformation φ: [0, ∞) → [0, ∞) that preserves φ(0) = 0 and for which φ is subadditive, yielding ||v||_φ = φ(||v||). This includes the standard norm as a special case when φ(t) = t. Other constructions involve norms on direct sums, or incorporating directional weights in the norm function to form weighted normstos, which can reflect anisotropic or problem-specific geometry.

Examples and applications: Standard p-norms can be viewed as specific normstos, as can weighted variants used in anisotropic spaces. Normstos appear in optimization, numerical analysis, and metric geometry, where flexibility in the triangle inequality or scaling behavior helps tailor distance measures to the structure of a given problem.

See also: Norm, P-norm, Minkowski functional.