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.
Graphs and maps with closed graphs
editIf is a map between topological spaces then the graph of
is the set
or equivalently,
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 graph
editIf 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 topology
editIn 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.)
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 functions
editClosed 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 analysis
editIf 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 also
edit- Almost open linear map – Map that satisfies a condition similar to that of being an open map.
- Barrelled space – Type of topological vector space
- Closed graph – Graph of a map closed in the product space
- Closed linear operator – Graph of a map closed in the product space
- Discontinuous linear map
- Kakutani fixed-point theorem – Fixed-point theorem for set-valued functions
- Open mapping theorem (functional analysis) – Condition for a linear operator to be open
- Ursescu theorem – Generalization of closed graph, open mapping, and uniform boundedness theorem
- Webbed space – Space where open mapping and closed graph theorems hold
- Zariski's main theorem – Theorem of algebraic geometry and commutative algebra
Notes
editReferences
edit- ^ Rudin 1991, p. 51-52.
- ^ Rudin 1991, p. 50.
- ^ Narici & Beckenstein 2011, pp. 459–483.
- ^ Munkres 2000, pp. 163–172.
- ^ Aliprantis, Charlambos; Kim C. Border (1999). "Chapter 17". Infinite Dimensional Analysis: A Hitchhiker's Guide (3rd ed.). Springer.
- ^ Schaefer & Wolff 1999, p. 78.
- ^ Trèves (2006), p. 173
Bibliography
edit- Bourbaki, Nicolas (1987) [1981]. Topological Vector Spaces: Chapters 1–5. Éléments de mathématique. Translated by Eggleston, H.G.; Madan, S. Berlin New York: Springer-Verlag. ISBN 3-540-13627-4. OCLC 17499190.
- Folland, Gerald B. (1984), Real Analysis: Modern Techniques and Their Applications (1st ed.), John Wiley & Sons, ISBN 978-0-471-80958-6
- Jarchow, Hans (1981). Locally convex spaces. Stuttgart: B.G. Teubner. ISBN 978-3-519-02224-4. OCLC 8210342.
- Köthe, Gottfried (1983) [1969]. Topological Vector Spaces I. Grundlehren der mathematischen Wissenschaften. Vol. 159. Translated by Garling, D.J.H. New York: Springer Science & Business Media. ISBN 978-3-642-64988-2. MR 0248498. OCLC 840293704.
- Munkres, James R. (2000). Topology (Second ed.). Upper Saddle River, NJ: Prentice Hall, Inc. ISBN 978-0-13-181629-9. OCLC 42683260.
- Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN 978-1584888666. OCLC 144216834.
- Rudin, Walter (1991). Functional Analysis. International Series in Pure and Applied Mathematics. Vol. 8 (Second ed.). New York, NY: McGraw-Hill Science/Engineering/Math. ISBN 978-0-07-054236-5. OCLC 21163277.
- Schaefer, Helmut H.; Wolff, Manfred P. (1999). Topological Vector Spaces. GTM. Vol. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135.
- Trèves, François (2006) [1967]. Topological Vector Spaces, Distributions and Kernels. Mineola, N.Y.: Dover Publications. ISBN 978-0-486-45352-1. OCLC 853623322.
- Wilansky, Albert (2013). Modern Methods in Topological Vector Spaces. Mineola, New York: Dover Publications, Inc. ISBN 978-0-486-49353-4. OCLC 849801114.
- Zălinescu, Constantin (30 July 2002). Convex Analysis in General Vector Spaces. River Edge, N.J. London: World Scientific Publishing. ISBN 978-981-4488-15-0. MR 1921556. OCLC 285163112 – via Internet Archive.
- "Proof of closed graph theorem". PlanetMath.