9
$\begingroup$

Let $\Vert f \Vert = |f(0)| + \mathrm{Var}f$ for all $f \in BV([0,1])$; we are given that it is a norm. Show that $BV([0,1])$ is a complete normed space with this norm.

I have shown that any Cauchy sequence in $BV([0,1])$ must converge to some function pointwise, but I am stuck at proving that the function must have bounded variation.

Could someone help me?

  • 1
    I think you can show that if $u_n$ is a sequence of $BV~$ functions that converge pointwise to $u$, then $\mathrm{Var}(u)\leq\mathrm{liminf} ~ \mathrm{Var}(u_n)\in\mathbb{R}_+\cup\lbrace\infty\rbrace$. Since you have a Cauchy sequence, the right hand side is finite, and this shows that the limit function has bounded variation.2011-10-17
  • 0
    I did prove that $\mathrm{Var}(u) \leq \liminf \mathrm{Var}(u_n)$ a few days ago, but I don't see how the sequence being Cauchy implies the right hand side is finite.2011-10-17
  • 5
    Cauchy sequences are **bounded**, thus for some $C>0$ and for all $n$, $$\mathrm{Var}(u_n)\leq ||u_n||\leq C.$$2011-10-17
  • 2
    Beware, your definition of $||\cdot||$ is missing vertical bars around $f$ : It should read $$||f||=|f(0)|+\mathrm{Var}(f).$$2011-10-17
  • 0
    Of course! I have been staring at this for so long that I neglected the simplest of tests! Thanks so much.2011-10-18
  • 0
    @nullUser Why don't you write an answer to your question?2012-06-23
  • 0
    but, how did you prove that $\|f-f_n\|\to 0$ ?2012-10-07

1 Answers 1

7

Let $\{f_n\}_{n=1}^{ +\infty} $ be a Cauchy sequence for $\lVert\cdot\rVert$. In particular, the sequence of real numbers $\{f_n(0)\}$ is Cauchy, hence converges to a real number we call $f(0)$. Now, considering the partition $t_0=0<1=t_1$, we have $$\lVert f_k-f_j\rVert\geqslant\operatorname{Var}(f_k-f_j)\geqslant \left|f_k(1)-f_j(1)\right|-\left|f_k(0)-f_j(0)\right|,$$ proving that $\left\{f_k(1)\right\}$ is Cauchy, hence converges to a real number called $f(1)$. Now for $t\in(0,1)$, we consider the partition $t_0:=0

  • $f$ is of bounded variation. Indeed, let $t_0=0

  • $\lVert f-f_N\rVert\to 0$. We have by definition $f_n(0)\to f(0)$ so we have to show that $\operatorname{Var}(f_n-f)\to 0$. Let $\varepsilon>0$. We can find $N=N(\varepsilon)$ such that if $m,n\geqslant N$ and $0=t_0