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I know the proof of the Schauder fixed point theorem which states

Schauder fixed point theorem : If $D$ is a non-empty , convex and compact subset of Banach space $B$ and $T:D \to D$ a continuous function then $T$ has a fixed point in $D$.

Now my question is how I can prove the Leray-Schauder fixed point theorem, which states

Leray-Schauder fixed point theorem : If $D$ is a non-empty , convex , bounded and closed subset of Banach space $B$ and $T:D \to D$ a compact and continuous map , then $T$ has a fixed point in $D$.

Can we prove the Leray-Schauder fixed point theorem with the Schauder fixed point theorem or are the proofs technically different?

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1 Answer 1

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$D$ is closed and bounded, and $T$ compact, hence $K = \overline{T(D)} \subset D$ is compact. Hence the convex hull $\operatorname{co} K$ is totally bounded, and $C = \overline{\operatorname{co} K} \subset D$ is a compact convex nonempty set. The restriction $T\lvert_C \colon C \to C$ is continuous. By the Schauder fixed point theorem, $T\lvert_C$ has a fixed point in $C$.

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