# Exterior algebra for infinite dimensional vector spaces

Let $$V$$ be a vector space. We know we have the associated exterior algebra which is $$\bigwedge V\equiv\bigoplus_{n=0}^{\infty}\bigwedge^nV$$ where $$\bigwedge^nV$$ is the $$n$$th exterior power of $$V$$.

If $$(e_j)_{j=1}^{\dim{V}}$$ is a basis for $$V$$, then we know $$(e_{j_1}\wedge\dots\wedge e_{j_n})_{ 1\leq j_1<\dots is a basis for $$\bigwedge^nV$$.

If we anti-symmetrize such elements: $$\sum_{\sigma\in S_n} \mathrm{sign}(\sigma)e_{j_1}\wedge\dots\wedge e_{j_n}$$ we get the so-called Slater determinant associated with $$j_1,\dots,j_n$$, which does not care about the ordering of the indices (via anti-symmetry) and so the various Slater determinants form the so-called "occupation basis" which is labelled not by a sequence $$j_1,\dots,j_n$$ but rather by stating whether a state $$j\in\{1,\dots,\dim{V}\}$$ appears in the determinant or not, i.e., specifying a sequence $$l_1,\dots,l_d\subseteq\{0,1\}$$ such that $$l_1+\dots+l_d=n$$.

Once we have the occupation basis though, we can forget about the constraint $$l_1+\dots+l_d=n$$ that places us in a particular direct summand $$\bigwedge^nV$$ and instead we may just think of any sequence $$l_1,\dots,l_d\subseteq\{0,1\}$$ as specifying a basis vector of $$\bigwedge V$$, and hence we have set up an isomorphism $$\bigwedge V\cong\bigotimes_{k=1}^{\dim V}\mathbb{C}^2\,.$$

My question is: does any of this make sense when $$V$$ is infinite dimensional? How to make sense of $$\bigotimes_{k=1}^{\infty}\mathbb{C}^2$$? I suppose one has to ask for some topological properties on $$V$$, e.g., that it be a Hilbert space? Given a Hilbert space $$\mathcal{H}$$, is there a canonical name for $$\bigotimes_{v\in B}\mathbb{C}^2$$ where $$B$$ is a basis of $$\mathcal{H}$$? Is that object indeed isomorphic to $$\bigwedge\mathcal{H}$$?

I don't see why the usual algebraic construction of the exterior algebra shouldn't go through even if the dimension of the underlying space $$V$$ is infinite. This goes by taking the free vector space on all elements of $$V$$ and then quotienting by the ideal generated by the exterior relation.