Closed graph theorem

The closed graph theorem can be reformulated as follows.

This follows from the closed graph theorem.

The open mapping theorem and closed graph theorem.

The closed graph theorem can be generalized to more abstract topological vector spaces in the following way:

In fact, the closed graph theorem, the open mapping theorem and the bounded inverse theorem are all equivalent.

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

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

The Baire category theorem about complete metric spaces, and its consequences, such as the open mapping theorem and the closed graph theorem.

However, boundedness of the inverse does follow directly from its existence if one introduces the additional assumption that T is closed; this follows from the closed graph theorem.

In mathematics, the closed graph theorem is a basic result in functional analysis which characterizes continuous linear operators between Banach spaces in terms of the operator graph.

Several important tools of functional analysis which are based on the Baire category theorem remain true in Fréchet spaces; examples are the closed graph theorem and the open mapping theorem.

For general functions, the graphic representation cannot be applied and the formal definition of the graph of a function suits the need of mathematical statements, e.g., the closed graph theorem in functional analysis.

The closed graph theorem asserts that all everywhere-defined closed operators on a complete domain are continuous, so in the context of discontinuous closed operators, one must allow for operators which are not defined everywhere.

It states that a bijective bounded linear operator T from one Banach space to another has bounded inverse T. It is equivalent to both the open mapping theorem and the closed graph theorem.

In fact, several principal results in functional analysis fail to hold in general for topological vector spaces: the closed graph theorem, the open mapping theorem, and the fact that the dual space of the space separates points in the space.

This is the content of the closed graph theorem: if the graph of T is closed in X x Y (with the product topology), we say that T is a closed operator, and, in this setting, we may conclude that T is continuous.