Search results
Results from the WOW.Com Content Network
Let : a function between topological vector spaces is said to be a locally bounded function if every point of has a neighborhood whose image under is bounded. The following theorem relates local boundedness of functions with the local boundedness of topological vector spaces:
Properties of a point on a function [ edit ] Perhaps the best-known example of the idea of locality lies in the concept of local minimum (or local maximum ), which is a point in a function whose functional value is the smallest (resp., largest) within an immediate neighborhood of points. [ 1 ]
By considering the cases of axioms 1 and 2 in which the multiset X has two elements and the case of axiom 3 in which the multisets X, Y, and Z have one element each, one recovers the usual axioms for a metric. That is, every multiset metric yields an ordinary metric when restricted to sets of two elements.
In mathematics, the Poincaré metric, named after Henri Poincaré, is the metric tensor describing a two-dimensional surface of constant negative curvature.It is the natural metric commonly used in a variety of calculations in hyperbolic geometry or Riemann surfaces.
Pavel Urysohn. In topology, the Tietze extension theorem (also known as the Tietze–Urysohn–Brouwer extension theorem or Urysohn-Brouwer lemma [1]) states that any real-valued, continuous function on a closed subset of a normal topological space can be extended to the entire space, preserving boundedness if necessary.
A map is a function, as in the association of any of the four colored shapes in X to its color in Y. In mathematics, a map or mapping is a function in its general sense. [1] These terms may have originated as from the process of making a geographical map: mapping the Earth surface to a sheet of paper. [2]
[1] The above construction can be seen as embedding a pointed metric space into a Banach space. The Kuratowski–Wojdysławski theorem states that every bounded metric space X is isometric to a closed subset of a convex subset of some Banach space. [2] (N.B. the image of this embedding is closed in the convex subset, not necessarily in the ...
Closed graph theorem for set-valued functions [6] — For a Hausdorff compact range space , a set-valued function : has a closed graph if and only if it is upper hemicontinuous and F(x) is a closed set for all .