If V and W are vector spaces of uncountably infinite dimension, they still have bases (according to axiom of choice).
Let basis sets be $\{v_x\}_{x \in X}$ and $\{w_y\}_{y \in Y}$, and define a set of linear transformations $\{l_{x, y}\}$ by $l_{x, y}(v_{x`}) = w_y$ if $x = x'$ and $=0$ otherwise.
I have only been able to find references to the use of integers or rationals for the Kronecker Delta. Is it a valid use of the Kronecker Delta to say $l_{x, y}(v_{x`}) = \delta_{x, x'}w_y$, and are there any references for this ? As a secondary question, if this is not valid, why not and what should one use instead ?
(For context, the set $\{l_{x, y}\}$ is linearly independent in $L(V, W)$ and is a basis if $V$ is finite dimensional and not a basis if V is infinite dimensional).