MHB Is $M$ a Finite Metric Space if $BC(M)$ is a Finite Real Vector Space?

[Euge]

Here is this week's POTW:

-----
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.-----

Remember to read the https://mathhelpboards.com/showthread.php?772-Problem-of-the-Week-%28POTW%29-Procedure-and-Guidelines to find out how to https://mathhelpboards.com/forms.php?do=form&fid=2!

 

[Euge]

No one answered this week's problem. You can read my solution below.

Spoiler

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.