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How can we prove that $(C_{00},\|\cdot\|)_{\ell^2}$ is not a Banach space?

How can we find counter example for this problem?

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    $\begingroup$ Consider the sequence $(x_n)$ with $x_n=\sum_{i=1}^n {1\over i}e_i$. $\endgroup$ Commented Jul 2, 2014 at 12:59
  • $\begingroup$ can you explain it please? $\endgroup$
    – Analysis
    Commented Jul 2, 2014 at 13:02
  • $\begingroup$ $(x_n)$ is Cauchy. But, it's not convergent: suppose $x_n\rightarrow x\in c_{00}$. Choose $n$ such that the $n$'th coordinate of $x$ is $0$. Then show $\Vert x_m-x\Vert \ge 1/n$ for all $m\ge n$. $\endgroup$ Commented Jul 2, 2014 at 13:09
  • $\begingroup$ I could not catch. how could you define the sequence what is e_i? $\endgroup$
    – Analysis
    Commented Jul 2, 2014 at 13:19
  • $\begingroup$ we have to take x_{n} from c_{00} dont we? $\endgroup$
    – Analysis
    Commented Jul 2, 2014 at 13:21

1 Answer 1

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Let me prove a stronger claim: $c_{00}$ is not a Banach space under any norm. Indeed, we can write

$$c_{00} = \bigcup_{n=1}^\infty\mbox{span}\{e_1, \ldots, e_n\}$$

where $e_n$ denotes the sequence which is everywhere null except the $n$th place where it takes value 1. No matter what norm we put on $c_{00}$ the spaces $\mbox{span}\{e_1, \ldots, e_n\}$ will be always closed because they are finite-dimensional. Now, if $c_{00}$ were a Banach space, by Baire's theorem, one of the subspaces $\mbox{span}\{e_1, \ldots, e_n\}$ would have non-empty interior. This is impossible as proper subspaces always have empty interior.

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  • $\begingroup$ This is a very nice, simple and elegant application of Baire Categories. Suddenly, I am able to construct an infinity of incomplete spaces!!! Thank you! I will take note! :-) $\endgroup$ Commented May 19, 2017 at 12:12

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