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Suppose $\mathbb{F}$ is a field, V and W are (possibly infinite-dimensional) vector spaces over $\mathbb{F}$. I denote their dual spaces as $V', W'$. Suppose $A$ and $B$ are (Hamel) bases of $V$ and $W$, respectively, while $C$ and $D$ are (Hamel) bases of $V'$ and $W'$, respectively. Can a basis of the vector space $\mathcal{L}(V, W)$ of linear maps from $V$ to $W$ be constructed in terms of $A, B, C, D$? I am fine with assuming the Axiom of Choice if it's needed.

I am asking this because I want to learn how tensor product works, and I like the definition of tensor product in terms of basis and I dislike the other definitions of tensor product. And to grok that definition, it would be useful for me to know how the bases of all the relevant vector spaces are related to each other.

CrabMan
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    Maybe check out this: https://math.stackexchange.com/questions/1178004/basis-for-tensor-product-of-infinite-dimensional-vector-spaces. However, I would advice you to get use to the definition in terms of the universal property. It is I think fundamental to proof many properties of tensor products. Once you get used to it, it will be way easier than dealing with those hamel bases. IMO, of course. – hal4math Apr 06 '21 at 22:54
  • Note that for infinite dimensional $V$, the dimension of $V^$ (the dual of $V$) is strictly larger than the dimension of $V$. That means that you are going to have issues in trying to use bases of $V^$: they are not as simple as the "dual bases" in the finite dimensional case, which correspond to bases of $V$. – Arturo Magidin Apr 06 '21 at 23:33

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