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Wednesday, March 9, 2016

Not a complete metric space

Prove that the set of the continuous functions $C[a, b]$ endowed with the metric

$$\rho(f, g) = \sqrt{\int_{a}^{b}\left | f(t)-g(t) \right |^2 \, {\rm d}t}$$

where $f, g \in C[a, b]$ is not a complete metric space.

Solution

Well, let us pick

$$f_n(x)=\left\{\begin{matrix}
0 & , & x \in \left[0, \frac{1}{2}\right] \\
 1& , &x \in \left [ \frac{1}{2}+ \frac{1}{n}, 1 \right ]
\end{matrix}\right.$$

and a line that connects $0$ with $1$ in that little segment. We easily note that $f_n$ are Cauchy because they are bounded and differ only by $1/n$ , but don't converge to a continuous function.

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