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# 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*

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