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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 wo$u$ld work. Will the Weierstrass approximation theorem give yo$u$ the result that sequence of partial sums of the sine series or cosine series constitute a sequence of polynomials not converging to a polynomial??2012-10-25

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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.