Induced Absolute Value
Given a locally compact topological field K, an absolute value can be defined as follows. First, consider the additive group of the field. As a locally compact topological group, it has a unique (up to positive scalar multiple) Haar measure μ. The absolute value is defined so as to measure the change in size of a set after multiplying it by an element of K. Specifically, define |·| : K → R by
for any measurable subset X of K (with 0 < μ(X) < ∞). This absolute value does not depend on X nor on the choice of Haar measure (since the same scalar multiple ambiguity will occur in both the numerator and the denominator).
Given such an absolute value on K, a new induced topology can be defined on K. This topology is the same as the original topology. Explicitly, for a positive real number m, define the subset Bm of K by
Then, the Bm make up a neighbourhood basis of 0 in K.
Read more about this topic: Local Field
Famous quotes containing the words absolute value, induced and/or absolute:
“One may almost doubt if the wisest man has learned anything of absolute value by living.”
—Henry David Thoreau (1817–1862)
“It is a misfortune that necessity has induced men to accord greater license to this formidable engine, in order to obtain liberty, than can be borne with less important objects in view; for the press, like fire, is an excellent servant, but a terrible master.”
—James Fenimore Cooper (1789–1851)
“José’s my first non-rat romance. Not that he’s my idea of the absolute finito. He’s too prim and cautious to be my absolute ideal, Now, if I could choose from anybody alive, I wouldn’t pick José. Nehru, maybe, or Albert Schweitzer. Or Leonard Bernstein.”
—George Axelrod (b. 1922)