In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, the closed graph theorem may refer to one of several basic results characterizing
continuous function
In mathematics, a continuous function is a function such that a continuous variation (that is a change without jump) of the argument induces a continuous variation of the value of the function. This means that there are no abrupt changes in value ...
s in terms of their
graph
Graph may refer to:
Mathematics
*Graph (discrete mathematics), a structure made of vertices and edges
**Graph theory, the study of such graphs and their properties
*Graph (topology), a topological space resembling a graph in the sense of discre ...
s.
Each gives conditions when functions with
closed graph
In mathematics, particularly in functional analysis and topology, closed graph is a property of functions.
A function between topological spaces has a closed graph if its graph is a closed subset of the product space .
A related property is o ...
s are necessarily continuous.
Graphs and maps with closed graphs
If
is a map between
topological space
In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called points ...
s then the graph of
is the set
or equivalently,
It is said that the graph of
is closed if
is a
closed subset
In geometry, topology, and related branches of mathematics, a closed set is a set whose complement is an open set. In a topological space, a closed set can be defined as a set which contains all its limit points. In a complete metric space, a clo ...
of
(with the
product topology
In topology and related areas of mathematics, a product space is the Cartesian product of a family of topological spaces equipped with a natural topology called the product topology. This topology differs from another, perhaps more natural-seemin ...
).
Any continuous function into a
Hausdorff space
In topology and related branches of mathematics, a Hausdorff space ( , ), separated space or T2 space is a topological space where, for any two distinct points, there exist neighbourhoods of each which are disjoint from each other. Of the many ...
has a closed graph.
Any linear map,
between two topological vector spaces whose topologies are (Cauchy) complete with respect to translation invariant metrics, and if in addition (1a)
is sequentially continuous in the sense of the product topology, then the map
is continuous and its graph, , is necessarily closed. Conversely, if
is such a linear map with, in place of (1a), the graph of
is (1b) known to be closed in the Cartesian product space
, then
is continuous and therefore necessarily sequentially continuous.
Examples of continuous maps that do ''not'' have a closed graph
If
is any space then the identity map
is continuous but its graph, which is the diagonal
, is closed in
if and only if
is Hausdorff. In particular, if
is not Hausdorff then
is continuous but does have a closed graph.
Let
denote the real numbers
with the usual
Euclidean topology
In mathematics, and especially general topology, the Euclidean topology is the natural topology induced on n-dimensional Euclidean space \R^n by the Euclidean distance, Euclidean metric.
Definition
The Euclidean norm on \R^n is the non-negative f ...
and let
denote
with the
indiscrete topology In topology, a topological space with the trivial topology is one where the only open sets are the empty set and the entire space. Such spaces are commonly called indiscrete, anti-discrete, concrete or codiscrete. Intuitively, this has the conseque ...
(where note that
is Hausdorff and that every function valued in
is continuous). Let
be defined by
and
for all
. Then
is continuous but its graph is closed in
.
Closed graph theorem in point-set topology
In
point-set topology
In mathematics, general topology is the branch of topology that deals with the basic set-theoretic definitions and constructions used in topology. It is the foundation of most other branches of topology, including differential topology, geomet ...
, the closed graph theorem states the following:
Non-Hausdorff spaces are rarely seen, but non-compact spaces are common. An example of non-compact
is the real line, which allows the discontinuous function with closed graph
.
For set-valued functions
In functional analysis
If
is a linear operator between
topological vector space
In mathematics, a topological vector space (also called a linear topological space and commonly abbreviated TVS or t.v.s.) is one of the basic structures investigated in functional analysis.
A topological vector space is a vector space that is als ...
s (TVSs) then we say that
is a
closed operator In mathematics, more specifically functional analysis and operator theory, the notion of unbounded operator provides an abstract framework for dealing with differential operators, unbounded observables in quantum mechanics, and other cases.
The ter ...
if the graph of
is closed in
when
is endowed with the product topology.
The closed graph theorem is an important result in functional analysis that guarantees that a closed linear operator is continuous under certain conditions.
The original result has been generalized many times.
A well known version of the closed graph theorems is the following.
See also
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Notes
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{{TopologicalVectorSpaces
Theorems in functional analysis