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Let $X$ be an infinite dimensional space over a field $F$ and $B$ a basis of $X$. We define two norms in $X$

$||x||_1=\sum_{i=1}^n|k_i|$ and $||x||_2=max\{|k_1|...|k_n|\}$ $\forall x \in X$,where $x=\sum_{i=1}^nk_ib_i$ and $b_1...b_n \in B$

Prove that these two norms are not equivalent.

Can someone help me with this or give me a hint?

Thank you in advance!

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  • $\begingroup$ Is $F=\mathbb{R}$ or $\mathbb{C}$ ? $\endgroup$ – Thibaut Dumont Feb 14 '17 at 22:52
  • $\begingroup$ @joeb, actually his base is in the sense that the $F$-span of $B$ is $X$. Hence every $x$ is a finite linear combination. $\endgroup$ – Thibaut Dumont Feb 14 '17 at 22:53
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Since $X$ is infinite dimensional, $B$ is infinite. Take a sequence $b_1,b_2,\ldots$ of independent vectors in $B$ and define $x_n=\sum_{i=1}^n b_i$. What are the norms of $x_n$ in both case? What happens when $n\to \infty$?

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