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Let $$X=\left(\begin{array}{c} x_1 \\ x_2\\ \vdots \end{array}\right)$$ be an infinite real vector and $$A=(a_{ij}), \ 0<i,j<\infty$$ be an infinite real matrix.

(1) For which $A$ can one define multiplication $AX$ on the space $\mathbb{R}^\infty$ of all infinite real vectors?

(2) What if we restrict $X$ to $Z=\{ X\in \mathbb{R}^\infty| x_n=0 \text{ for all but finitely many }n\}$

My answer is

(1) $A$ must be such that for all $i$, $a_{ij}=0$ for all but finitely many $j$.

(2) It is well defined for all $A=(a_{ij}), \ 0<i,j<\infty$

Is that correct?

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  • $\begingroup$ In (1) - is with "all infinite real vectors" meant: any arbitrary vector X ? And why/for what purpose in this question A is a matrix and not only a row-vector? $\endgroup$ – Gottfried Helms Jun 2 '16 at 4:44
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Basically yes, you can define an infinite matrix-times-column product provided each matrix row is finitely supported, or (without restricting the matrix) when the column vector is finitely supported.

Of course with "can one define" questions, it will sometimes be possible to stretch the domain of application by restricting what one requires from a definition. In this case one could in principle allow some cases to be defined provided all the infinite sums that arise in the usual definition converge in some sense. I'm not sure how many useful properties the result would have; in any case it is not in the spirit of abstract (exact) algebra to let analysis thus penetrate into the basic definitions. (But of course, one can look at things like Hilbert spaces where such things can be done meaningfully.)

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