Prove that there are $2$ independent vectors $x,y \in(l_\infty)$ such that $||x||=||y||=1$ and $||x+y||=2$
(Here $l_\infty = \{(a_n)|a_n\in \mathbb{C}, sup|a_n|<\infty\}$ and $||x||=sup|x_n|$).
I was thinking about the following -
$x=(x_n)=(1-\frac{1}{n})$, $y=(y_n)=1$.
It is easy to see that $||x||=||y||=1$, and also that $||x+y||=||1-\frac{1}{n}+1||=2$.
What I'm not sure is, are $x,y$ independent?
(My definition of being independent - $x_1,x_2,...,x_n\in E$ are independent in $E_0$ if: $\sum_{i=1}^n\alpha_ix_i \in E_0 \iff\alpha_i=0 \forall 1\le i \le n$).