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.

单词	SchauderFixedPointTheorem
释义	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.
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