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$a)$ $C^1[0,1]$ of continuously differentiable real valued functions on $[0,1]$ with the metric $$d(f,g)=\max_{t\in[0,1]}|f-g|$$

I am sure that it is not complete, but could any one help me to construct a counter example? well, $f(x)=|x-1/2|$ will work?

$b)$ The space of all polynomial with real coeffi single variable with same metric as above, this is also not complete, $|x-1/2|$ will work?

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For the 2nd one..I was thinking whether this would work. Will the Weierstrass approximation theorem give you the result that sequence of partial sums of the sine series or cosine series constitute a sequence of polynomials not converging to a polynomial?? – Vishesh Oct 25 '12 at 16:01
up vote 2 down vote accepted

For part (a), your function $f$ is not differentiable at $x=1/2$, and so $f\not\in C^1[0,1]$.

To get a counterexample, try to find a limiting function $f\in C^0[0,1]\setminus C^1[0,1]$ (i.e. a continuous, non-differentiable function) and a sequence of smooth functions that converges to $f$ in $C^0[0,1]$. This convergent (in $C^0$) sequence can be chosen to give you a Cauchy sequence in $C^1[0,1]$ with the metric you have indicated - but the limit 'falls out' of $C^1$, giving the counterexample you need. Indeed your suggested $f$ could be used as the limiting function. For the sequence, apply a mollifier to $f$.

Similarly for (b), your suggested $f$ is not in the space in question.

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