In the following, the norm of a normed algebraic structure (group, left module, vector space) is defined by the (given) distance function (the norm of an element is its distance from the zero element, see nmval: ). By this definition, the norm function is actually a norm (satisfying the properties: being a function into the reals; subadditivity/triangle inequality ; absolute homogeneity ( n(sx) = |s| n(x) ) [Remark: for group norms, some authors (e.g., Juris Steprans in "A characterization of free abelian groups") demand that n(sx) = |s| n(x) for all , whereas other authors (e.g., N. H. Bingham and A. J. Ostaszewski in "Normed versus topological groups: Dichotomy and duality") only require inversion symmetry, i.e., . The definition df-ngp of a group norm follows the second approach, see nminv.] and positive definiteness/point-separation ) if the structure is a metric space with a right-translation-invariant metric (see nmf, nmtri, nmvs and nmeq0). An alternate definition of a normed group (i.e., a group equipped with a norm) not using the properties of a metric space is given by Theorem tngngp3. The norm can be expressed as the distance to zero (nmfval), so in a structure with a zero (a "pointed set", for instance a monoid or a group), the norm can be expressed as the distance restricted to the elements of the base set to zero (nmfval0).
Usually, however, the norm of a normed structure is given, and the corresponding metric ("induced metric") is defined as the distance function based on the norm (the distance between two elements is the norm of their difference, see ngpds: ). The operation does exactly this, i.e., it adds a distance function (and a topology) to a structure (which should be at least a group for the difference of two elements to make sense) corresponding to a given norm as explained above: , see also tngds. By this, the enhanced structure becomes a normed structure if the induced metric is in fact a metric (see tngngp2) or a norm (see tngngpd). If the norm is derived from a given metric, as done with df-nm, the induced metric is the original metric restricted to the base set: , see nrmtngdist, and the norm remains the same: , see nrmtngnrm.