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

the closed graph theorem is a result connecting the continuity of a linear operator to a topological property of their graph. Precisely, the theorem states...

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

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

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

unbounded operator. The closed graph theorem says a linear operator f : X → Y {\displaystyle f:X\to Y} between Banach spaces is a closed operator if and only...

graph of Γ {\displaystyle \Gamma } is the set of all a ∈ A {\displaystyle a\in A} such that Γ ( a ) {\displaystyle \Gamma (a)} is not empty. Theorem—If...

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

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

redirect targets Closed graph theorem – Theorem relating continuity to graphs Closed graph theorem (functional analysis) – Theorems connecting continuity...

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...

Alspach's theorem (graph theory) Aztec diamond theorem (combinatorics) BEST theorem (graph theory) Baranyai's theorem (combinatorics) Berge's theorem (graph theory)...

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...

a graph coloring of the planar graph of adjacencies between regions. In graph-theoretic terms, the theorem states that for a loopless planar graph G {\displaystyle...

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...

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...

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...

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...

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...

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

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...

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...

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

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

tangent (as in the case of the absolute value represented in the graph). Rolle's theorem implies that a differentiable function whose derivative is ⁠ 0...

spaces, the closed range theorem gives necessary and sufficient conditions for a closed densely defined operator to have closed range. The theorem was proved...

Frucht's theorem is a result in algebraic graph theory, conjectured by Dénes Kőnig in 1936 and proved by Robert Frucht in 1939. It states that every finite...

graph introduced by Shannon. The conjecture remained unresolved for 40 years, until it was established as the celebrated strong perfect graph theorem...

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