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HandWiki. Closed Graph. Encyclopedia. Available online: https://encyclopedia.pub/entry/29880 (accessed on 12 October 2026).
HandWiki. Closed Graph. Encyclopedia. Available at: https://encyclopedia.pub/entry/29880. Accessed October 12, 2026.
HandWiki. "Closed Graph" Encyclopedia, https://encyclopedia.pub/entry/29880 (accessed October 12, 2026).
HandWiki. (2022, October 18). Closed Graph. In Encyclopedia. https://encyclopedia.pub/entry/29880
HandWiki. "Closed Graph." Encyclopedia. Web. 18 October, 2022.
Closed Graph
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The content is sourced from: https://handwiki.org/wiki/Closed_graph

In mathematics, particularly in functional analysis and topology, closed graph is a property of functions. A function f : X → Y between topological spaces has a closed graph if its graph is a closed subset of the product space X × Y. A related property is open graph. This property is studied because there are many theorems, known as closed graph theorems, giving conditions under which a function with a closed graph is necessarily continuous. One particularly well-known class of closed graph theorems are the closed graph theorems in functional analysis.

functional analysis topological spaces topology

References

  1. Narici & Beckenstein 2011, pp. 459-483.
  2. Munkres 2000, p. 171.
  3. Aliprantis, Charlambos; Kim C. Border (1999). "Chapter 17". Infinite Dimensional Analysis: A Hitchhiker's Guide (3rd ed.). Springer. 
  4. Rudin p.50
  5. Narici & Beckenstein 2011, p. 480.
  6. Kreyszig, Erwin (1978). Introductory Functional Analysis With Applications. USA: John Wiley & Sons. Inc.. pp. 294. ISBN 0-471-50731-8. 
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