# Right multiplication on $\mathbb{R}^{\infty}$

If $X=(x_1,x_2,\dots)$ is an infinite real row vector and $A=(a_{ij}),0<i,j< \infty$ is an infinite real matrix, one may or may not be able to define the matrix product $XA$. For which A can one define right multiplication on the space $\mathbb{R}^\infty$ of all infinite row vectors ? on the space $Z$ defined as $Z=\{(a) \in \mathbb{R}^{\infty}|a_n=0 \text{for all but finitely many n} \}$

$\textbf{My attempt: }$ $X$ is an infinite Real Row matrix such as, $X = (x_1, x_2, \dots) = (x_n)$ where $n = 1, 2,\dots$ And $A$ is also an infinite real matrix such as, $A = (a_{ij})$ We now pass to the most general matrix-by-matrix product, and consider the operations involved in computing the product $C$ of two matrices $A$ and $X$: $C = XA$

Let $A=(a_{ij})_{ij}$ be the matrix of order $i \times j$ where $0<i$ and $j< \infty$, $X=(x_1 \cdots x_n)$ be the row matrix of order $1 \times n$ and $C=(c_{ik})_{ik}$ is possible only when i is the same for both infinite matrices.

$C=XA=\sum_{j=1}^n x_na_{ij}$

This sum is defined only if $n=i$

Therefore, one may define the product $XA$ over infinite space $\mathbb{R}^{\infty}$ given that the above condition is satisfied.

Is it correct? How do I proceed on Z?

• Could someone help me with the Z case ? – user43418 Oct 10 '13 at 17:51

Given $X,A$ we can define the product if for every $i$, the series $\sum_jx_ja_{ji}$ converges. If we have to define the product for all $X$, then it follows that the matrices $A$ should satisfy the criteria that for all $i$ there exists $N$ such that $a_{ji}=0$ for $j>N$.
To multiply $X$ and $A$, the sum $x_1 a_{1j} + x_2 a_{2j} + \dots$ must have only finitely many nonzero terms. In order for this to be true for all row vectors $X$, the column $(a_{ij}, a_{2j}, \dots)$ must have only finitely many entries different from zero. So $A$ must be a column-finite matrix.
To multiply when $X \in Z$, i.e., $X$ has finitely many nonzero elements, we can use an arbitrary matrix $A$. However, if we want the answer $XA$ to be an element of $Z$ for every $X \in Z$, then the rows of $A$ must have finitely many entries different from zero: $A$ must be a row-finite matrix. This is seen by trying $X = e_i$.