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Show that the metric $d_1$ on $C([0,1])$ does not give rise to a complete metric space.

$$ d_1(f,g) = \int_0^1 |f(s)−g(s)| \, ds $$

Andrés E. Caicedo
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Tom
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    Think about the metric. What do you think the completion of the space will be? Consider the functions $f_{n}(x) = x^n$ and $g(x) = 0 \forall x \in [0, 1)$ and $g(1) = 1$. Why might this work? See @RonMor's response on how you might fill in the details. – Christopher K Nov 01 '14 at 18:13
  • Welcome to MSE. Please let us know what work you've done towards and answer and where you are stuck. –  Nov 01 '14 at 18:22

1 Answers1

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What have you tried so far?

My recommendation:

  1. find a Cauchy's series $f_n$ that you think "should" converge to some function $f$, but $f \not\in C([0,1])$.

  2. Define $K=C([0,1])\cup${$f$}

  3. show that $d_1$ is still a metric, and $\lim f_n = f$

  4. Conclude by the uniqueness of the limit, there is no limit for $f_n$ in $C([0,1])$, hence it is not complete.

If you won't be able to find such series I'll try to help you with that.

Best of luck

  • Yeah, that's definitely the structure I need to follow for my solution.

    I have already been trying to find a series like that but I'm just struggling with where to start finding an example.

    Thanks for the help!

     – Tom Nov 01 '14 at 18:23