$\ell^1$ is not complete for the norm $\|\cdot\|_\infty$
Solution 1:
For the first one, if $m>n$ $$ \|u^{(n)}-u^{(m)}\|_\infty=\sup\{(0,\ldots,0,\frac1{n+1},\frac1{n+2},\ldots,\frac1m)\}=\frac1{n+1}. $$ So the sequence is Cauchy.
For the second one, let $$u^{(n)}=v^{(n)}=\bigg(\underbrace{\frac1{\sqrt n},\ldots,\frac1{\sqrt n}}_{n \text{ times }},0,\ldots\bigg).$$ Then $\|u^{(n)}\|_\infty=\|v^{(n)}\|_\infty=\frac1{\sqrt n}\to0$, while $\phi(u^{(n)},v^{(n)})=1$ for all $n$.