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:
“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)
“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)
“... if, as women, we accept a philosophy of history that asserts that women are by definition assimilated into the male universal, that we can understand our past through a male lens—if we are unaware that women even have a history—we live our lives similarly unanchored, drifting in response to a veering wind of myth and bias.”
—Adrienne Rich (b. 1929)