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Recall the following theorem:

Theorem: If $X$ is Banach space and $A$ is a bounded linear map on $X$ with $\|A\| < 1$, then $\sum\limits_{i=1}^{\infty}A^{i}$ converges, and the limit is $(I-A)^{-1}$.

I am trying to thinking of a counter example in case $X$ is not Banach.

That is, I want to find a normed vector space $Y$ that is not Banach, and a linear map $B$ on $Y$ such that $\|B\| < 1$ but $\sum\limits_{i=1}^{\infty}B^i$ does not exist as a bounded linear map on $B$.

The non-banach space that I feel one could find an example in is the space $Y = \ell_{c}(\mathbb{N})$, which consists of of infinite sequences with only finitely many non-zero entries, together with the norm $\|a\| =\sum\limits_{i=1}^{\infty} |a_i|$.

Can I find a map $A$ on this space with $\|A\| <1$ but such that the series $\sum\limits_{i=1}^{\infty}A^i$ does not converge to a bounded linear map on the space?

Thanks!

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    $\begingroup$ One problem that would occur for non-complete spaces is simply that a Cauchy sequence of vectors need not have a limit in the space. But/and this is not a pathology. $\endgroup$
    – paul garrett
    Jun 23, 2017 at 1:53
  • $\begingroup$ Hi, I'm familiar with examples of Cauchy sequences without limits in non-complete normed spaces (such as $\ell_{c}(\mathbb{N})$. Could you give me a hint on how I could explicitly construct a linear map $A$ with the desired properties? $\endgroup$
    – ttb
    Jun 23, 2017 at 2:15
  • $\begingroup$ btw, I still have in my drawer print outs of your abstract algebra notes that I studied from a few years back! $\endgroup$
    – ttb
    Jun 23, 2017 at 2:22
  • $\begingroup$ For $Ae_n=e_{n+1}/2$ on the finite vectors in $\ell^2$, your sum is $\sum e_n/2^n$, which converges in $\ell^2$ of course, but has infinitely-many non-zero components. (Algebra notes! :) $\endgroup$
    – paul garrett
    Jun 23, 2017 at 17:22

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Try the composition of shift and division by 2: $$A(a_1,a_2,a_3,\dots) = (0,a_1/2,a_2/2,a_3/2,\dots)$$

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