Given the normed space $C^1[0,1]$ of differentiable functions with continuous derivatives on $[0,1]$. The norm is defined as
$$\|x\| = \max_{[0,1]} |x(t)|$$
I'd like to prove that the given normed space is not a Banach space.
In the attempt of solving this problem, I have thought about the possibility to construct a Cauchy sequence in the space which doesn't converge in the space. However until now I haven't got any idea. Then I proceeded to think about constructing an equivalent norm to the given one where the space can be easily proven to be not a Banach space. However, I have got nothing either.
Now I'm stuck without a clue. Please give me a hint to a correct direction.
Any help is greatly appreciated.
Best Answer
Take $\sqrt{x+1/n}$, which is $C^1[0,1]$ for each $n$ thanks to our shifting of the discontinuity in the derivative.
To see this converges uniformly, note that $\sqrt{x}$ is uniformly continuous on compact sets. On say $[0,2]$ we may use uniform continuity to meet any epsilon challenge with an $N$ not dependent on $x$ with $$ |\sqrt{x+1/N}-\sqrt{x}|<\epsilon $$