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YCor
YCor
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I deleted a sentence which was obvious by the compactness.
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Sh.M1972
Sh.M1972
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I am not an expert of this area but I need help to answer this question: For a compact metric space $X$, let $C(X,X)$ be the space of all continuous functions $X\to X$, equipped with the uniform metric $$d(f,g)=\sup_x d(f(x),g(x)).$$ Can we say the space $C(X,X)$ is compact? In the special case which I need my space $X$ is also complete and separable.

I am not an expert of this area but I need help to answer this question: For a compact metric space $X$, let $C(X,X)$ be the space of all continuous functions $X\to X$, equipped with the uniform metric $$d(f,g)=\sup_x d(f(x),g(x)).$$ Can we say the space $C(X,X)$ is compact? In the special case which I need my space $X$ is also complete and separable.

I am not an expert of this area but I need help to answer this question: For a compact metric space $X$, let $C(X,X)$ be the space of all continuous functions $X\to X$, equipped with the uniform metric $$d(f,g)=\sup_x d(f(x),g(x)).$$ Can we say the space $C(X,X)$ is compact?

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Sh.M1972
Sh.M1972
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Compactness of $C(X,X)$

I am not an expert of this area but I need help to answer this question: For a compact metric space $X$, let $C(X,X)$ be the space of all continuous functions $X\to X$, equipped with the uniform metric $$d(f,g)=\sup_x d(f(x),g(x)).$$ Can we say the space $C(X,X)$ is compact? In the special case which I need my space $X$ is also complete and separable.