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Prove that the space $C[0,1]$ of continuous functions from $[0,1]$ to $\mathbb{R}$ with the inner product $ \langle f,g \rangle =\int_{0}^{1} f(t)g(t)dt \quad $ is not Hilbert space.

I know that I have to find a Cauchy sequence $(f_n)_n$ which converges to a function $f$ which is not continuous, but I can't construct such a sequence $(f_n)_n$.

Any help?

  • 4
    Please, oh please, write the inner product using `\langle f,g\rangle`, resulting in $\langle f,g\rangle$. `<` and `>` are for inequalities.2012-02-22
  • 1
    Try to find a sequence converging to a step function with just two values.2012-02-22
  • 0
    @HaraldHanche-Olsen That's one of my pet peeves. I've even had professors that wrote $(\cdot,\cdot)$ for the inner product. How hard is it to use the correct, unambiguous notation?2016-05-06

2 Answers 2