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Then f : X → Y is continuous but its graph is not closed in X × Y. [4] If X is any space then the identity map Id : X → X is continuous but its graph, which is the diagonal Gr Id := { (x, x) : x ∈ X }, is closed in X × X if and only if X is Hausdorff. [7] In particular, if X is not Hausdorff then Id : X → X is continuous but not closed.
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).
Precisely, the theorem states that a linear operator between two Banach spaces is continuous if and only if the graph of the operator is closed (such an operator is called a closed linear operator; see also closed graph property). An important question in functional analysis is whether a given linear operator is continuous (or bounded).
A real function that is a function from real numbers to real numbers can be represented by a graph in the Cartesian plane; such a function is continuous if, roughly speaking, the graph is a single unbroken curve whose domain is the entire real line. A more mathematically rigorous definition is given below. [8]
The fundamental group can be defined for discrete structures too. In particular, consider a connected graph G = (V, E), with a designated vertex v 0 in V. The loops in G are the cycles that start and end at v 0. [4] Let T be a spanning tree of G. Every simple loop in G contains exactly one edge in E \ T; every loop in G is a concatenation of ...
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).
If G has C components (disconnected graphs), the same argument by induction on F shows that + = . One of the few graph theory papers of Cauchy also proves this result. Via stereographic projection the plane maps to the 2-sphere, such that a connected graph maps to a polygonal decomposition of the sphere, which has Euler characteristic 2.
Let X, Y be Banach spaces. An unbounded operator (or simply operator) T : D(T) → Y is a linear map T from a linear subspace D(T) ⊆ X —the domain of T —to the space Y. [5] Contrary to the usual convention, T may not be defined on the whole space X. An operator T is said to be closed if its graph Γ(T) is a closed set. [6]