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# Tychonoff fixed point theorem

Let $X$ be a locally convex topological 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 a normed vector space is a locally convex topological vector space so this theorem extends the Schauder fixed point theorem.

# References

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

Related:

SchauderFixedPointTheorem, BrouwerFixedPointTheorem

Type of Math Object:

Theorem

Major Section:

Reference

## Mathematics Subject Classification

54H25*no label found*46B50

*no label found*47H10

*no label found*

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