Let $S \subset C^2[0,1]$ (set of two times differentiable functions $f(x)$ on $[0,1]$) which satisfy the following: $$\int_0^1 f(x)\,dx\leq3$$ Question is $(S,d)$ is a complete metric space, where $$d(f,g)=\sup_{x \in [0,1]}|f(x)-g(x)|+ \sup_{x \in [0,1]}|f'(x)-g'(x)| + \sup_{x \in [0,1]}|f''(x)-g''(x)|.$$
I am somewhat sure that I can prove that $d$ is a metric indeed. The rest, a complete mystery. I will almost definitely attach a boundy to this question.