Cauchy sequences of compactly supported functions

Question

We have the three standard norms $\left\Vert\right\Vert_1$, $\left\Vert\right\Vert_2$, $\left\Vert\right\Vert_\infty$ defined on the space of compactly supported functions $C_0(\Bbb N,\Bbb R)$, $f: \Bbb N \rightarrow \Bbb R$.

For each pair of norms, I must to find sequences which are cauchy with respect to one of these norms but not the other. For instance, in considering the $\left\Vert\right\Vert_1$ and $\left\Vert\right\Vert_2$ norms, I must find a sequence of functions that is cauchy for $\left\Vert\right\Vert_1$ but not $\left\Vert\right\Vert_2$, for the $\left\Vert\right\Vert_1$ and $\left\Vert\right\Vert_\infty$ norms I must to find a sequence of functions that is cauchy for $\left\Vert\right\Vert_1$ but not $\left\Vert\right\Vert_\infty$, and for the $\left\Vert\right\Vert_2$ and $\left\Vert\right\Vert_\infty$ norms I must find a sequence of functions that is cauchy for $\left\Vert\right\Vert_2$ but not $\left\Vert\right\Vert_\infty$ or vice versa.

I have attempted many approaches but I am having difficulty building an intuition for the behavior of these norms in the context of sequences of functions. Any guidance would be much appreciated.

Answer 1

Note that $\Vert \cdot \Vert_\infty \leqslant \Vert \cdot \Vert _1$ and $\Vert \cdot \Vert_\infty \leqslant \sqrt{\Vert \cdot \Vert _2}$ and when $\Vert f \Vert_1 <1$, then $\Vert f \Vert_2 \leqslant \Vert f \Vert_1$. So any Cauchy sequence for $\Vert \cdot \Vert _1$ or for $\Vert \cdot \Vert _2$ will be a Cauchy sequence for $\Vert \cdot \Vert _\infty$, and any Cauchy sequence for $\Vert \cdot \Vert _1$ will be a Cauchy sequence for $\Vert \cdot \Vert _2$

Check the inequalities and this will suppress many cases for which there is a Cauchy sequence for a norm but not for another.

For the other cases, basically use the sequences of sequences $$(f_N)_{N\in \mathbb{N}} \text{ with } f_N=(1,1/2,\cdots,1/N,0,0,0,\cdots)$$ and $$(g_N)_{N\in \mathbb{N}} \text{ with } g_N=(1,1/\sqrt{2},\cdots,1/\sqrt{N},0,0,0,\cdots).$$

Answer 2

Hint: Let $E_k=\{1,\dots ,k\}$. Then let $f_{k}(n):=\frac{1}{n}1_E(n)$. So for example, $f_1(1)=1$, $f_1(n)=0$ for $n>1$. Then $f_2(1)=1$, $f_2(2)=\frac{1}{2}$, $f_2(n)=0$ for $n>2$. Then $f_3(1)=1$, $f_3(2)=\frac{1}{2}$, $f_3(3)=\frac{1}{3}$, $f_3(n)=0$ for $n>3$. And so on. Now let $f(n):=\frac{1}{n}$. What types of convergences are happening?