Closed graph theorem

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

A cubic function
The Heaviside function
The graph of the cubic function on the interval is closed because the function is continuous. The graph of the Heaviside function on is not closed, because the function is not continuous.

Graphs and maps with closed graphsEdit

If   is a map between topological spaces then the graph of   is the set   or equivalently,

 
It is said that the graph of   is closed if   is a closed subset of   (with the product topology).

Any continuous function into a Hausdorff space 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, Gr L, 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.[1]

Examples of continuous maps that do not have a closed graphEdit

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.[2] In particular, if   is not Hausdorff then   is continuous but does not have a closed graph.

Let   denote the real numbers   with the usual Euclidean topology and let   denote   with the indiscrete topology (where note that   is not Hausdorff and that every function valued in   is continuous). Let   be defined by   and   for all  . Then   is continuous but its graph is not closed in  .[3]

Closed graph theorem in point-set topologyEdit

In point-set topology, the closed graph theorem states the following:

Closed graph theorem[4] — 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   is compact. (Note that compactness and Hausdorffness do not imply each other.)

Proof

First part is essentially by definition.

Second part:

For any open   , we check   is open. So take any   , we construct some open neighborhood   of   , such that   .

Since the graph of   is closed, for every point   on the "vertical line at x", with   , draw an open rectangle   disjoint from the graph of   . These open rectangles, when projected to the y-axis, cover the y-axis except at   , so add one more set  .

Naively attempting to take   would construct a set containing  , but it is not guaranteed to be open, so we use compactness here.

Since   is compact, we can take a finite open covering of   as  .

Now take  . It is an open neighborhood of  , since it is merely a finite intersection. We claim this is the open neighborhood of   that we want.

Suppose not, then there is some unruly   such that   , then that would imply   for some   by open covering, but then   , a contradiction since it is supposed to be disjoint from the graph of   .

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 functionsEdit

Closed graph theorem for set-valued functions[5] — For a Hausdorff compact range space  , a set-valued function   has a closed graph if and only if it is upper hemicontinuous and F(x) is a closed set for all  .

In functional analysisEdit

If   is a linear operator between topological vector spaces (TVSs) then we say that   is a closed operator 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.

Theorem[6][7] — A linear map between two F-spaces (e.g. Banach spaces) is continuous if and only if its graph is closed.

See alsoEdit

NotesEdit

ReferencesEdit

  1. ^ Rudin 1991, p. 51-52.
  2. ^ Rudin 1991, p. 50.
  3. ^ Narici & Beckenstein 2011, pp. 459–483.
  4. ^ Munkres 2000, pp. 163–172.
  5. ^ Aliprantis, Charlambos; Kim C. Border (1999). "Chapter 17". Infinite Dimensional Analysis: A Hitchhiker's Guide (3rd ed.). Springer.
  6. ^ Schaefer & Wolff 1999, p. 78.
  7. ^ Trèves (2006), p. 173

BibliographyEdit