Norm (mathematics) - Properties

Properties

The concept of unit circle (the set of all vectors of norm 1) is different in different norms: for the 1-norm the unit circle in R2 is a square, for the 2-norm (Euclidean norm) it is the well-known unit circle, while for the infinity norm it is a different square. For any p-norm it is a superellipse (with congruent axes). See the accompanying illustration. Note that due to the definition of the norm, the unit circle is always convex and centrally symmetric (therefore, for example, the unit ball may be a rectangle but cannot be a triangle).

In terms of the vector space, the seminorm defines a topology on the space, and this is a Hausdorff topology precisely when the seminorm can distinguish between distinct vectors, which is again equivalent to the seminorm being a norm. The topology thus defined (by either a norm or a seminorm) can be understood either in terms of sequences or open sets. A sequence of vectors is said to converge in norm to if as . Equivalently, the topology consists of all sets that can be represented as a union of open balls.

Two norms ||•||_α and ||•||_β on a vector space V are called equivalent if there exist positive real numbers C and D such that

for all x in V. For instance, on, if p > r > 0, then

In particular,

If the vector space is a finite-dimensional real/complex one, all norms are equivalent. On the other hand, in the case of infinite-dimensional vector spaces, not all norms are equivalent.

Equivalent norms define the same notions of continuity and convergence and for many purposes do not need to be distinguished. To be more precise the uniform structure defined by equivalent norms on the vector space is uniformly isomorphic.

Every (semi)-norm is a sublinear function, which implies that every norm is a convex function. As a result, finding a global optimum of a norm-based objective function is often tractable.

Given a finite family of seminorms p_i on a vector space the sum

is again a seminorm.

For any norm p on a vector space V, we have that for all u and v ∈ V:

p(u ± v) ≥ | p(u) − p(v) |

For the lp norms, we have Hölder's inequality

A special case of this is the Cauchy–Schwarz inequality: