Definition
Given a vector space V over a subfield F of the complex numbers, a norm on V is a function p: V → R with the following properties:
For all a ∈ F and all u, v ∈ V,
- p(av) = |a| p(v), (positive homogeneity or positive scalability).
- p(u + v) ≤ p(u) + p(v) (triangle inequality or subadditivity).
- If p(v) = 0 then v is the zero vector (separates points).
A simple consequence of the first two axioms, positive homogeneity and the triangle inequality, is p(0) = 0 and thus
- p(v) ≥ 0 (positivity).
A seminorm is a norm with the 3rd property (separating points) removed.
Every vector space V with seminorm p(v) induces a normed space V/W, called the quotient space, where W is the subspace of V consisting of all vectors v in V with p(v) = 0. The induced norm on V/W is clearly well-defined and is given by:
- p(W + v) = p(v).
A topological vector space is called normable (seminormable) if the topology of the space can be induced by a norm (seminorm).
Read more about this topic: Norm (mathematics)
Famous quotes containing the word definition:
“It’s a rare parent who can see his or her child clearly and objectively. At a school board meeting I attended . . . the only definition of a gifted child on which everyone in the audience could agree was “mine.””
—Jane Adams (20th century)
“Although there is no universal agreement as to a definition of life, its biological manifestations are generally considered to be organization, metabolism, growth, irritability, adaptation, and reproduction.”
—The Columbia Encyclopedia, Fifth Edition, the first sentence of the article on “life” (based on wording in the First Edition, 1935)
“Perhaps the best definition of progress would be the continuing efforts of men and women to narrow the gap between the convenience of the powers that be and the unwritten charter.”
—Nadine Gordimer (b. 1923)