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.
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.