Prove that the space $C^1([a,b])$ consisting of continuous functions in $[a, b]$ with the norm $|x(a)| + \max \{|x'(t)|: t \in [a,b]\}$ is a Banach space.
I can't prove the completeness of this space. Hope someone can help me. Thanks. I really appreciate this.
That's not even a norm. Take $a = 0$, $b=1$, $x(t) = -2 + 2t$. Did you mean "$|x(a)|$"? By the way, "\max" gives you $\max$ and is easier on the eyes.
EDIT: after I wrote the above, the question was fixed and "$x(a)$" was replaced by "$|x(a)|$", so the given norm (I'll call it "$\| \cdot \|_l$") actually is a norm. The norm $\| \cdot \|_l$ is equivalent to the standard norm on $C^1([a,b])$, which is $\|f\| = \max_{[0,1]} |f| + \max_{[0,1]} |f'|$: to verify this you must verify two inequalities, one of which is trivial, and the other one of which takes a little work. Then a sequence of functions $(f_n) \subset C^1([0,1])$ is Cauchy in one of the norms if and only if it is Cauchy in the other norm. If you are allowed to assume $C^1([0,1])$ is complete using the standard norm, it should be easy to prove $C^1([0,1])$ is complete using $\| \cdot \|_l$.