# Infinite matrix product

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 $Z=\{ X\in \mathbb{R}^\infty| x_n=0 \text{ for all but finitely many }n\}$

(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?

• 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? – Gottfried Helms Jun 2 '16 at 4:44