I am trying to construct a sequence of functions in $C[0,1]$ with the $L^{1/2}$ metric $$d(f, g) = \int_0^1 \sqrt{|f(x) - g(x)|} \,dx$$ that is Cauchy but not convergent.
There are a whole bunch of questions on the site with examples for $L^1$ (though they're all basically the same), and I've been trying to adapt them for this case. Unfortunately, no matter how I tweak my sequence it makes calculating the actual distance $d(f_n,f_m)$ (and showing that this goes to $0$, etc.) very difficult, because of the square root.
Does anyone have a concrete and reasonably computable example to show that $C[0,1]$ is not complete with the $L^{1/2}$ metric? Your help is appreciated.
For $n=1,2,\dots,$ define $f_n(x)= 1/x, 1/n\le x \le 1,$ $f_n(x)= n, 0\le x \le 1/n.$ This is a sequence in $C[0,1]$ that converges to $1/x$ in $L^{1/2}.$