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The normed vector space in question is $(C^{1}[0,1],\Vert \cdot \Vert _c )$ where $\| f \| _c := \lvert f(1) \rvert + \lVert f' \rVert _{1}$

For the first part of the question I had to work out whether or not $\| \cdot \| _c $ was a norm, which I believe it is. The second part was to work out which norms made the above vector space a complete normed space (there were norms indexed with $a$ and $b$ but this is the one I really need help with) i.e is $(C^{1}[0,1],\| \cdot \| _c )$ complete?

I feel as though it isn't complete, my intuition is that simply evaluating $f(x)$ at $1$ isn't enough 'information' or perhaps it's better to say that it's not restrictive enough? However, I'm really bad at giving counterexamples in these types of questions, I was thinking of constructing a Cauchy sequence $(f_n)$ that converges to a continuous function with a discontinuous derivative and set $f_n(1)=0$ which would simplify things.

Any ideas of a counterexample? or maybe an explanation as to why it maybe complete? Thank you.

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HINT: Do you know any example of a Cauchy sequence in $(C[0,1],\|\cdot\|_1)$ that does not converge? By means of integration you might be able to turn this into an example of a Cauchy sequence in $(C^1[0,1],\|\cdot\|_c)$ that does not converge.

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  • $\begingroup$ Thanks for the response, I do actually have such an example in my lecture notes. A sequence of functions where identically $0$ for $x \in [0,1/2]$ then a straight line for $x \in [1/2,1/2 +1/n]$ then identically $1$ for the rest of domain. This converge to a step function, so are you saying find a function who's derivative behaves like this? $\endgroup$ – Displayname Mar 2 '19 at 12:07
  • $\begingroup$ Of course these functions You describe are not in $C^1([0,1])$. Do You know mollifiers?@Displayname $\endgroup$ – Peter Melech Mar 2 '19 at 12:10
  • $\begingroup$ @Displayname That is exactly what I was aiming at. With the fundamental theorem of calculus you should be able to finish. $\endgroup$ – SmileyCraft Mar 2 '19 at 12:11
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    $\begingroup$ Not sure if this works, if You take the functions You describe You'll get something like $$f(x)=\begin{cases}C_1,0\leq x\leq \frac{1}{2}\\ \frac{n}{2}x^2-\frac{n}{2}x+C_2,\frac{1}{2}\leq x\leq \frac{1}{2}+\frac{1}{n}\\x+C_3,\frac{1}{2}+\frac{1}{n}\leq x\leq 1\end{cases}$$ and You are right, these are in $C^1$ $\endgroup$ – Peter Melech Mar 2 '19 at 12:37
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    $\begingroup$ for appropriate constants $\endgroup$ – Peter Melech Mar 2 '19 at 12:43

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