For each $ n\ge 1$ Define the vectors $e\_n = (e\_{nk})$ where $ k\ge 1$ and $ e\_{nk} = \frac{1}{k^n}$
Is this set a basis for $l^2$?
Thanks,



Determining if a set is a Basis for l^2
You want to know if $\sum\_{k=1}^\infty a\_k k^{-n}=0$ for $(a\_k)\in l^2$ and every positive integer $n$ implies $a\_k=0$. This is true. First we note that if $(a\_k)\in l^2$, then $(a\_k/k)\in l^1$. So w.l.o.g. we may consider the case where $(a\_k)\in l^1$ to begin with. Now note that the condition implies that $\sum\_{k=1}^\infty a\_kp(1/k)=0$ for every polynomial $p$ with $p(0)=0$. Now use the Weierstrass approximation theorem to conclude that $\sum\_{k=1}^\infty a\_k f(1/k)=0$ for every continuous function $f$ on $[0,1]$ with $f$ vanishing near the origin. It follows easily that $a\_k=0$.


