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Consider the vector space, $V$ of doubly infinite sequences of complex numbers over the complex field. Let $e_i$ denote the sequence which is 1 at the integer $i$ and $0$ elsewhere. I am looking for a linear transformation $T$ from $V$ to $V$ and a $ x = (x_i)$, such that $T(x) = \sum_{i} x_i T(e_i)$ is "invalid" (i.e., some coordinate value on the RHS does not converge to the corresponding value on the LHS).

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    Let $x$ be the all-ones sequence, and let $T$ be any linear map which sends every $e_i$ to $x$ itself (there are many such maps, because the set $\{e_i\}$ does not span $V$)2012-04-01
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    Thanks a lot, is it possible to find such a $T$ explicitly. The only argument I can think of is that the $e_i$'s are contained in some basis. I would prefer an explicit map since I need to explain the problem with the infinite sum to someone with an engineering background.2012-04-01
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    You are not going to be able to explicit exhibit a basis of that vector space to an engineer :)2012-04-01

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