Matrix-like Representation of any linear map using Hamel Basis Let $X$ and $Y$ be two arbitrary linear spaces over a field $\mathbb{K}$. Let $B=\{x_\alpha:\alpha \in S\}$ and $C=\{y_\beta:\beta \in T\}$ be Hamel basis for $X$ and $Y$ respectively. Denote the space of all the linear maps from $X$ into $Y$ by  $L(X,Y)$. For any set $Z$, consider $$C_{00}(Z)=\{(d_\gamma)_{\gamma\in Z}\in\mathbb{K}^Z:d_\gamma\neq0 \mbox{ only for a finite number of indexes }\gamma\}.$$
Let $F\in L(X,Y)$ and $x\in X$. Then there exist a unique $a=(a_\alpha)\in C_{00}(S)$ such that $x=\sum_\alpha a_\alpha x_\alpha$. 
Also, for each $\alpha\in S$, there is a unique $M_\alpha=(b_{\beta,\alpha})_\beta\in C_{00}(T)$ such that $F(x_\alpha)=\sum_\beta b_{\beta,\alpha}y_\beta$. 
Thus 
$$F(x)=\sum_\alpha a_\alpha F(x_\alpha)=\sum_\alpha a_\alpha\bigg(\sum_\beta b_{\beta,\alpha}y_\beta\bigg)=\sum_\beta \bigg(\sum_\alpha b_{\beta,\alpha}a_\alpha\bigg)y_\beta$$
Consider the "matrix-like array $M$ of size $|T|\times|S|$" which columns are the $|T|$-tuples $M_\alpha$.
Thus $M\in(C_{00}(T))^S$.
 Now, for each $\beta\in T$, let $N_\beta$ be the "$\beta$-row of $M$", i.e. $N_\beta=(b_{\beta,\alpha})_{\alpha\in S}$. 
If we denote $\sum_\alpha b_{\beta,\alpha}a_\alpha$ by $\langle N_\beta,a\rangle$ for each $\beta \in T$, then
$$F(x)=\sum_\beta\langle N_\beta,a\rangle y_\beta.$$
Questions:


*

*I think the "matrix-like array $M$" is uniquely determined by the linear map $F$ and conversely. In other words $L(X,Y)\cong(C_{00}(T))^S.$ Am I right?

*Is this representation for linear maps of any use?

*In spaces like $\ell^p$, we don't have any explicit Hamel basis, so (I guess) this approach doesn't allow us to get information about the continuity of any operator from its respective array. Am I wrong?

 A: Here's the good news. This "matrix" form does represent the operator. Uniqueness of everything will suffice to show we do have an honest representation of your operator.
Here's the bad news. These things won't behave like matrices in any meaningful way (as far as I know). What does a transpose do? What do minors look like? Oh god what does a Jordan normal form mean? At the absolute best case (we're talking about miracles here), it looks like the kernel of an integral operator, and then you might as well use the theory of integral operators. 
Part of the problems with Hamel basis in general is clear from your work. All the notation is pretty bad. Yours is as clean as it gets, as far as I know. Maybe it looks a little better using cardinals instead of arbitrary index sets, but it doesn't really solve the problem. Certainly you are quite right that the lack of easily defined Hamel basis make this hard to study. And that Hamel basis for Banach spaces are always uncountable is problematic as well. 
All that being said, I think this is great. Nothing would make me happier than a new approach to Banach space geometry. As nerdy as that sounds, I'm remarkably serious. Please keep working on this if it interests you. 
If I were going to pursue this, I would look for two things:


*

*Basis independent properties. I cannot overstate the importance of this one. All your basis will be terrible. So try to build tools to help get around this fact. 

*Analogies from matrix theory. Some of these already exist in reasonable ways. Look up nuclear operators for an analogy of trace in infinite dimensions (I study these). Figure out what the diagonal should even mean in this context. Take stronger assumptions on $S$ and $T$ if you need them. Seriously, I think they should be cardinals.
A: *

*Right. 

*For infinite dimensional spaces this is not very useful with the exception that Hamel bases can be used to "construct" exotic things like additive functions from $\mathbb R \to \mathbb R$ which are not linear. (That's why Hamel invented Hamel bases.)

*For Banach spaces one can work with Schauder bases but then one has to be careful with convergence problems. For Hilbert spaces orthonormal Schauder bases are indeed very useful.
