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Closed graph theorem [5] — If : is a map from a topological space into a Hausdorff space, then the graph of is closed if : is continuous. The converse is true when Y {\displaystyle Y} is compact .
In mathematics, particularly in functional analysis, 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 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 ...
Closed graph theorems are of particular interest in functional analysis where there are many theorems giving conditions under which a linear map with a closed graph is necessarily continuous. If f : X → Y is a function between topological spaces whose graph is closed in X × Y and if Y is a compact space then f : X → Y is continuous.
A family of graphs is minor-closed if it is closed under minors; the 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 both directed and undirected edges. modular 1.
1. The closed graph theorem states that a linear operator between Banach spaces is continuous (bounded) if and only if it has closed graph. 2. A closed operator is a linear operator whose graph is closed. 3. The closed range theorem says that a densely defined closed operator has closed image (range) if and only if the transpose of it has ...
In functional analysis, a branch of mathematics, a closed linear operator or often a closed operator is a linear operator whose graph is closed (see closed graph property). It is a basic example of an unbounded operator. The closed graph theorem says a linear operator between Banach spaces is a closed operator if and only if it is a bounded ...
Closed graph theorem (functional analysis) Closed range theorem (functional analysis) Cluster decomposition theorem (quantum field theory) Coase theorem ; Cochran's theorem ; Codd's theorem (relational model) Cohen structure theorem (commutative algebra) Cohn's irreducibility criterion (polynomials)
Theorem — If : is an upper hemicontinuous set-valued function with closed domain (that is, the domain of is closed) and closed values (i.e. () is closed for all ), then is closed. If B {\displaystyle B} is compact, then the converse is also true.