Give an example of a linearly independent sequence $[x_0,x_1,x_2,\dots]$ of vectors in $\ell^{\infty}$ such that $\sum_{n=1}^{\infty} (x_n) =0$

Question

Give an example of a linearly independent sequence $[x_0,x_1,x_2,\dots]$ of vectors in $\ell^{\infty}$ such that $\sum_{n=1}^{\infty} (x_n) =0$.

Solution:

Let $X_n=[x_0,x_1,\dots]$; define $x_0=[-1,0,0,0,\dots]$, $x_1=[0,-1,0,0,\dots]$ and so on.

Let $S_n$ be a partial sums, $S_n=x_0+x_1+x_2+\dots+x_n$. Then $\|S_n\| \to 0$ as $n \to \infty$ where we defined $\|S_n\|=\sup|S_k|$.

Thank you for your suggestion. :')

Answer 1

Hint: You're missing something here. It seems that what you wanted to do was define $$ x_0 = [1,1,1,\dots] $$ Then, define $$ x_1 = -[1,0,0,\dots]\\ x_2 = -[0,1,0,\dots] $$ and so forth. Note, however, that this won't work, since we always have $\|S_n\| = 1$. On the other hand, we could begin by defining $$ x_0 = [1,1/2,1/3,\dots] $$ why does this work any better?