## You are here

HomeSchauder fixed point theorem

## Primary tabs

# Schauder fixed point theorem

Let $X$ be a normed vector space, and let $K\subset X$ be a non-empty, compact, and convex set. Then given any continuous mapping $f\colon K\to K$ there exists $x\in K$ such that $f(x)=x$.

Notice that the unit disc of a finite dimensional vector space is always convex and compact hence this theorem extends Brouwer Fixed Point Theorem.

Notice that the space $X$ is not required to be complete, however the subset $K$ being compact, is complete with respect to the metric induced by $X$.

# References

- 1
Rudin,
*Functional Analysis*, Chapter 5.

Related:

BrouwerFixedPointTheorem, FixedPoint, TychonoffFixedPointTheorem

Type of Math Object:

Theorem

Major Section:

Reference

## Mathematics Subject Classification

54H25*no label found*47H10

*no label found*46B50

*no label found*

- Forums
- Planetary Bugs
- HS/Secondary
- University/Tertiary
- Graduate/Advanced
- Industry/Practice
- Research Topics
- LaTeX help
- Math Comptetitions
- Math History
- Math Humor
- PlanetMath Comments
- PlanetMath System Updates and News
- PlanetMath help
- PlanetMath.ORG
- Strategic Communications Development
- The Math Pub
- Testing messages (ignore)

- Other useful stuff

## Recent Activity

Jul 5

new correction: Error in proof of Proposition 2 by alex2907

Jun 24

new question: A good question by Ron Castillo

Jun 23

new question: A trascendental number. by Ron Castillo

Jun 19

new question: Banach lattice valued Bochner integrals by math ias

new correction: Error in proof of Proposition 2 by alex2907

Jun 24

new question: A good question by Ron Castillo

Jun 23

new question: A trascendental number. by Ron Castillo

Jun 19

new question: Banach lattice valued Bochner integrals by math ias