extended norm

It is sometimes convenient to allow the norm to take extended real numbers as values. This way, one can accomdate elements of infiniteMathworldPlanetmath norm in one’s vector spaceMathworldPlanetmath. The formal definition is the same except that one must take care in stating the second condition to avoid the indeterminate form 0.

Definition: Given a real or complex vector space V, an extended norm is a map ¯ which staisfies the follwing three defining properties:

  1. 1.

    Positive definiteness: v>0 unless v=0, in which case v=0.

  2. 2.

    Homogeneity: λv=|λ|v for all non-zero scalars λ and all vectors v.

  3. 3.

    Triangle inequalityMathworldMathworldPlanetmathPlanetmath: u+vu+v for all u,vV.

Example Let C0 be the space of continuous functionsMathworldPlanetmathPlanetmath on the real line. Then the function defined as


is an extended norm. (We define the supremumMathworldPlanetmathPlanetmath of an unbounded set as .) The reason it is not a norm in the strict sense is that there exist continuous functions which are unbounded. Let us check that it satisfies the defining properties:

  1. 1.

    Because of the absolute valueMathworldPlanetmathPlanetmathPlanetmath in the definition, it is obvious that f0 for all f. Furthermore, if f=0, then supx|f(x)|=0, so |f(x)|0 for all x, which implies that f=0.

  2. 2.

    If f is boundedPlanetmathPlanetmathPlanetmathPlanetmath, then


    If f is unbounded and λ0, then λ|f| is unbounded as well. By the convention that infinityMathworldPlanetmath times a finite number is infinity, property (2) holds in this case as well.

  3. 3.

    If both f and g are bounded, then


    On the other hand, if either f or g is unbounded, then the right hand side of (3) is infinity by the convention that anything other minus infinity added to plus infinity is still infinity and infinity is bigger than anything which could appear on the left.

An extended norm partitionsMathworldPlanetmathPlanetmathPlanetmath a vector space into equivalence classesMathworldPlanetmathPlanetmath modulo the relationMathworldPlanetmath u-v<. The equivalence class of zero is a vector space consisting of all elemets of finite norm. Restricted to this equivalence class the extended norm reduces to an ordinary norm. The other equivalence classes are translatesMathworldPlanetmath of the equivalence class of zero. Furthermore, when we use the distance function of the norm to define a topologyMathworldPlanetmath on our vector space, these equivalence classes are exactly the connected componentsMathworldPlanetmathPlanetmathPlanetmathPlanetmath of the topology.

Title extended norm
Canonical name ExtendedNorm
Date of creation 2013-03-22 14:59:49
Last modified on 2013-03-22 14:59:49
Owner rspuzio (6075)
Last modified by rspuzio (6075)
Numerical id 6
Author rspuzio (6075)
Entry type Definition
Classification msc 46B99