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Closed graph theorem

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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.

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 T. Tao’s blog post[1] lists several closed graph theorems throughout mathematics.

Graphs and maps with closed graphs

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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 (see § Closed graph theorem in point-set topology)

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.[2]

Examples of continuous maps that do not have a closed graph

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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.[3] 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 .[4]

Closed graph theorem in point-set topology

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

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 is compact. (Note that compactness and Hausdorffness do not imply each other.)

Proof

First part: just note that the graph of is the same as the pre-image where is the diagonal in .

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 .

If X, Y are compact Hausdorff spaces, then the theorem can also be deduced from the open mapping theorem for such spaces; see § Relation to the open mapping theorem.

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 .

Also, closed linear operators in functional analysis (linear operators with closed graphs) are typically not continuous.

For set-valued functions

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Closed graph theorem for set-valued functions[6] — 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 analysis

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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[7][8] — A linear map between two F-spaces (e.g. Banach spaces) is continuous if and only if its graph is closed.

The theorem is a consequence of the open mapping theorem; see § Relation to the open mapping theorem below (conversely, the open mapping theorem in turn can be deduced from the closed graph theorem).

Relation to the open mapping theorem

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Often, the closed graph theorems are obtained as corollaries of the open mapping theorems in the following way.[1][9] Let be any map. Then it factors as

.

Now, is the inverse of the projection . So, if the open mapping theorem holds for ; i.e., is an open mapping, then is continuous and then is continuous (as the composition of continuous maps).

For example, the above argument applies if is a linear operator between Banach spaces with closed graph, or if is a map with closed graph between compact Hausdorff spaces.

See also

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Notes

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References

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  1. ^ a b https://terrytao.wordpress.com/2012/11/20/the-closed-graph-theorem-in-various-categories/
  2. ^ Rudin 1991, p. 51-52.
  3. ^ Rudin 1991, p. 50.
  4. ^ Narici & Beckenstein 2011, pp. 459–483.
  5. ^ Munkres 2000, pp. 163–172.
  6. ^ Aliprantis, Charlambos; Kim C. Border (1999). "Chapter 17". Infinite Dimensional Analysis: A Hitchhiker's Guide (3rd ed.). Springer.
  7. ^ Schaefer & Wolff 1999, p. 78.
  8. ^ Trèves (2006), p. 173
  9. ^ https://arxiv.org/abs/2403.03904

Bibliography

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