= Solution
For $x\ne y$, $|(\delta_x-\delta_y)(f)|\leq2\|f\|_\infty$, giving an upper bound $2$ for the <operator norm>. By <Urysohn lemma>, there is $h:K\to[0,1]$ with $h(x)=1,h(y)=0$. Then $f=2h-1$ has <uniform norm> $1$ and $(\delta_x-\delta_y)(f)=2$. Thus every distinct pair has <operator norm> $2$, and
$$
\boxed{\inf_{x\ne y}\|\delta_x-\delta_y\|=2\quad\text{if }|K|\geq2.}
$$
The stated hypotheses also permit a singleton $K$. Then there are no distinct pairs, so the infimum is \b[$+\infty$ under the extended-real convention], or undefined as a real number. This edge case cannot be assigned the value $2$.
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