Search results
Results From The WOW.Com Content Network
Given a function: from a set X (the domain) to a set Y (the codomain), the graph of the function is the set [4] = {(, ()):}, which is a subset of the Cartesian product.In the definition of a function in terms of set theory, it is common to identify a function with its graph, although, formally, a function is formed by the triple consisting of its domain, its codomain and its graph.
If the element y in Y is assigned to x in X by the function f, one says that f maps x to y, and this is commonly written = (). In this notation, x is the argument or variable of the function. A specific element x of X is a value of the variable , and the corresponding element of Y is the value of the function at x , or the image of x under the ...
set-valued function with a closed graph. If F : X → 2 Y is a set-valued function between topological spaces X and Y then the following are equivalent: F has a closed graph (in X × Y); (definition) the graph of F is a closed subset of X × Y; and if Y is compact and Hausdorff then we may add to this list:
Theorem [7] [8] — A linear map between two F-spaces (e.g. Banach spaces) is continuous if and only if its graph is closed. The theorem is a consequence of the open mapping theorem ; see § Relation to the open mapping theorem below (conversely, the open mapping theorem in turn can be deduced from the closed graph theorem).
A graph with three vertices and three edges. A graph (sometimes called an undirected graph to distinguish it from a directed graph, or a simple graph to distinguish it from a multigraph) [4] [5] is a pair G = (V, E), where V is a set whose elements are called vertices (singular: vertex), and E is a set of unordered pairs {,} of vertices, whose elements are called edges (sometimes links or lines).
The Borel graph theorem, proved by L. Schwartz, shows that the closed graph theorem is valid for linear maps defined on and valued in most spaces encountered in analysis. [10] Recall that a topological space is called a Polish space if it is a separable complete metrizable space and that a Souslin space is the continuous image of a Polish space ...
The points P 1, P 2, and P 3 (in blue) are collinear and belong to the graph of x 3 + 3 / 2 x 2 − 5 / 2 x + 5 / 4 . The points T 1, T 2, and T 3 (in red) are the intersections of the (dotted) tangent lines to the graph at these points with the graph itself. They are collinear too.
A graph function f is TG-invariant if: [2] = ... The numbers c, x, y, a, b are parameters. Generalization to matroids. The matroid function f is TG if: [1]