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Problem Of The Week # 347 - May 21, 2019

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Euge

MHB Global Moderator
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Jun 20, 2014
1,925
Here is this week's POTW:

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Prove that if $M$ is a metric space, then $M$ is finite if and only if the set $BC(M)$ of bounded continuous functions $f : M \to \Bbb R$ is a finite dimensional real vector space.


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Remember to read the POTW submission guidelines to find out how to submit your answers!
 
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Euge

MHB Global Moderator
Staff member
Jun 20, 2014
1,925
No one answered this week's problem. You can read my solution below.


The set of evaluation maps $\{\operatorname{ev}_x:x\in M\}$ is linearly independent in the dual space $BC(M)^*$, so $M$ has cardinality not exceeding the dimension of $BC(M)^*$; if $BC(M)$ is finite dimensional, so is $BC(M)*$, so then $M$ has finite cardinality, i.e., $M$ is finite. Conversely, if $M$ is finite, say, $M = \{x_1,\ldots, x_n\}$, the maps $f_i : M \to \Bbb R$ given by $f_i(x) = 1_{x_i}(x)$ are elements of $BC(M)$ and form a basis for $BC(M)$. Thus, $BC(M)$ is finite dimensional.
 
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