Many texts reads "It is well known that for infinite matrices multiplication is non-associative". A treatise on this can be found in On the associativity of infinite matrix multiplication.
However, if $x$ and $v$ are infinite vectors, and $A$ is a semi-infinite matrix, then Does the product $x^\intercal A u$ is associative?, that is to ask whether the following equation holds?
$$(x^\intercal A) u = x^\intercal (A u)$$
I am interested in the particular case when $A$ is a lower-triangular semi-infinite matrix.