Show that if $||\cdot||_1$, $||\cdot||_2$ are equivalent norms then $(V,||\cdot||_1)$ is a banach space iff $(V,||\cdot||_2)$ is.

Question

Show that if $||\cdot||_1$, $||\cdot||_2$ are equivalent norms then $(V,||\cdot||_1)$ is a banach space iff $(V,||\cdot||_2)$ is. I really didn't get it. Of course both spaces are normed spaces but there are two things I don't understand:

1. Why is it so important to look at "Cauchy sequences"? Why can't I look at converging sequences? Aren't those equivalent?

2. If those are equivalent, ones a sequence converges to a limit (which is in the space) should it be different in the other "space", that is $(V,||\cdot||_2)?

I really didn't understand what I need to do here. I would truly appreciate your help.

Answer 1

It suffices to show that

1. a sequence $\left\{v_n\right\}$ is Cauchy in $\left\Vert\cdot\right\Vert_1$ iff it is Cauchy in $\left\Vert\cdot\right\Vert_2$.

2. a sequence $\left\{v_n\right\}$ converges in $\left\Vert\cdot\right\Vert_1$ iff it converges in $\left\Vert\cdot\right\Vert_2$.

Why is this sufficient? Because then the argument goes like this: Suppose $\left\Vert\cdot\right\Vert_1$ is complete; that is, $V$ is a Banach space in the $\left\Vert\cdot\right\Vert_1$ metric. Let $\left\{v_n\right\}$ be a Cauchy sequence in $\left\Vert\cdot\right\Vert_2$. By (1.) the sequence is Cauchy in $\left\Vert\cdot\right\Vert_1$. Since $\left\Vert\cdot\right\Vert_1$ is complete, the sequence converges in $\left\Vert\cdot\right\Vert_1$, and by (2.) the sequence converges in $\left\Vert\cdot\right\Vert_2$. Therefore $\left\Vert\cdot\right\Vert_2$ is complete.

Answer 2

Because completeness means that all Cauchy sequences are convergent. Thus, to prove completeness obviously you have to take a Cauchy sequence and prove that it is convergent. This is so obvious that it's hard to understand your difficulty. Maybe you are used to Euclidean spaces, which are always complete, so there's nothing to prove in Euclidian space; but in infinite dimensional spaces you have to prove it.