dimensionnorms

Dimensionnorms refer to a family of norms or seminorms on a finite-dimensional vector space that encode contributions from different dimensions or coordinate subspaces. The term is not standardized in mathematical literature, but it appears in some applied contexts to describe dimension-aware measures of vector size or energy.

Formally, let V be a real or complex vector space of finite dimension n. For each d in {1, ..., n}, choose a linear map A_d: V → R^{m_d} (often a projection onto a fixed d-dimensional subspace). Define the d-th dimensionnorm by ||x||_d = ||A_d x||, where ||·|| on R^{m_d} is a chosen norm (for example the Euclidean norm). Then ||·||_d is a seminorm on V in general, and becomes a norm precisely when A_d is injective on V. This framework allows a dimension parameter to influence the measured size of a vector.

Examples: with V = R^n and A_d being the projection onto the first d coordinates, and ||·||_2 on R^d, one gets ||x||_d = sqrt(x_1^2 + ... + x_d^2). This is a seminorm on R^n, vanishing for vectors supported entirely on coordinates beyond d.

To obtain a genuine norm that reflects all coordinates, one may combine the dimensionnorms, e.g., ||x|| = (sum_{d=1}^n w_d ||A_d x||^p)^{1/p} with positive weights w_d and p ≥ 1, or take the maximum across d. Such constructions yield norms on V that still encode dimension-related information via the A_d.

Applications include analysis of dimensional decomposition, multi-subspace learning, and evaluating coordinate-wise contributions in data analysis. Because dimensionnorms are not a single standard object, conventions vary by field.

See also: norm, seminorm, projection, dimensionality reduction, subspace.