Let $(X,d)$ be a compact metric space and let $S= \{f \in C(X):\|f\|\le 1\}$ be the closed unit ball of $C(X)$. Show that if $X$ is an infinite set then $S$ will not be compact.
for infinite compact set $X$ the closed unit ball of $C(X)$ will not be compact
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analysis
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1http://en.wikipedia.org/wiki/Riesz'_lemma – 2012-10-04
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0Thanks Michael ,please let me know is this work for the metric space also? – 2012-10-04
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0The space of continuous functions on a compact metric space with the sup-norm is an infinite dimensional normed space. Finite dimensional subspaces are closed. You can use this to find a countable family of disjoint open balls with the same radius in the unit ball. By covering the rest with very small balls, you get an open cover without a countable subcover. – 2012-10-04
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0hmmm it seems to be use full,can you explain a bit more so that i can easily do my work on it? – 2012-10-05