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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.

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  • $\begingroup$ Note that the (completeness) tag was deemed to have no real value. See the meta discussion here. $\endgroup$
    – user642796
    Jul 30, 2015 at 18:54

1 Answer 1

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Hint: note that $C^2[0,1]$ is a complete space under the metric $d$. So, it suffices to show that if we have a sequence of functions $\{f_n\} \subset S$ with $f_n \to f$ in $C^2[0,1]$, then $f \in S$. That is, it suffices to prove the following result:

Suppose that $\{f_n\} \subset C^2[0,1]$ satisfies $\lim_{n \to \infty}d(f_n,f) = 0$ and that for each $n$, we have $$ \int_0^1 f_n(x)\,dx \leq 3 $$ Then $f$ satisfies $\int_0^1 f(x)\,dx \leq 3$.

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