mathematics, the closed graph theorem may refer to one of several basic results characterizing continuous functions in terms of their graphs. Each gives conditions...

topology, the closed graph theorem is a result connecting the continuity of certain kinds of functions to a topological property of their graph. In its most...

graph theory, the Robertson–Seymour theorem (also called the graph minor theorem) states that the undirected graphs, partially ordered by the graph minor...

function with a closed graph is necessarily continuous. One particularly well-known class of closed graph theorems are the closed graph theorems in functional...

major theorems which are sometimes called the four pillars of functional analysis: the Hahn–Banach theorem the open mapping theorem the closed graph theorem...

and convex analysis, the Ursescu theorem is a theorem that generalizes the closed graph theorem, the open mapping theorem, and the uniform boundedness principle...

whether any minor-closed class of graphs is determined by a finite set of "forbidden minors". This is now the Robertson–Seymour theorem, proved in a long...

closed graph theorem). Local convexity of X {\displaystyle X} or Y {\displaystyle Y}  is not essential to the proof, but completeness is: the theorem...

spaces) and φ is required to be closed-valued in the alternative statement of the Kakutani theorem, the Closed Graph Theorem implies that the two statements...

inverse T−1. It is equivalent to both the open mapping theorem and the closed graph theorem. Theorem — If A : X → Y is a continuous linear bijection from...

} has open lower sections then it is lower hemicontinuous. The open graph theorem says that if Γ : A → P ( R n ) {\displaystyle \Gamma :A\to P\left(\mathbb...

Ore's theorems basically state that a graph is Hamiltonian if it has enough edges. The Bondy–Chvátal theorem operates on the closure cl(G) of a graph G with...

complement of a graph hole. Chordless cycles may be used to characterize perfect graphs: by the strong perfect graph theorem, a graph is perfect if and...

to the graph of the function) is zero. The theorem is named after Michel Rolle. If a real-valued function f is continuous on a proper closed interval...

The theory of graph minors began with Wagner's theorem that a graph is planar if and only if its minors include neither the complete graph K5 nor the complete...

topology, the Jordan curve theorem, formulated by Camille Jordan in 1887, asserts that every Jordan curve (a plane simple closed curve) divides the plane...

Robertson–Seymour theorem characterizes minor-closed families as having a finite set of forbidden minors. mixed A mixed graph is a graph that may include...

2-factor theorem (graph theory) 15 and 290 theorems (number theory) 2π theorem (Riemannian geometry) AF+BG theorem (algebraic geometry) ATS theorem (number...

Borel graph theorem is generalization of the closed graph theorem that was proven by L. Schwartz. The Borel graph theorem shows that the closed graph theorem...

with the goal of allowing the results of the open mapping theorem and the closed graph theorem to hold for a wider class of linear maps whose codomains...

Otto Toeplitz. This theorem can be viewed as an immediate corollary of the closed graph theorem, as self-adjoint operators are closed. Alternatively, it...

planar graph can be formed from a map in this way. In graph-theoretic terminology, the four-color theorem states that the vertices of every planar graph can...

is used in functional analysis to prove the open mapping theorem, the closed graph theorem and the uniform boundedness principle. BCT1 also shows that...

subset. Here, I {\displaystyle I} is the identity operator. By the closed graph theorem, λ {\displaystyle \lambda } is in the spectrum if and only if the...

Hahn–Banach theorem may sometimes be used to show that an extension exists. However, the extension may not be unique. Closed graph theorem (functional...

forbidden graphs, the complete graph K5 and the complete bipartite graph K3,3. For Kuratowski's theorem, the notion of containment is that of graph homeomorphism...

the Blumberg theorem guarantees that even this function has some dense subset on which its restriction is continuous. Closed graph theorem (functional...

In graph theory, Wagner's theorem is a mathematical forbidden graph characterization of planar graphs, named after Klaus Wagner, stating that a finite...

theorem is a tool that allows relations to be converted to functions of several real variables. It does so by representing the relation as the graph of...

graph theory, graph coloring is a special case of graph labeling; it is an assignment of labels traditionally called "colors" to elements of a graph subject...