This question is related to my [prior question](https://mathoverflow.net/questions/363935/what-is-the-role-of-topology-on-infinite-dimensional-exterior-algebras), but this one is aimed, even though it's more general. If $V$ is a vector space, we define the exterior algebra of $V$ do be:
$$\bigwedge V := \bigoplus\_{n=0}^{\infty}\bigwedge^{n}V $$
where $\bigwedge^{n}V$ is the $n$-fold exterior power of $V$ and I used the identification $\bigwedge^{0}V = \mathbb{C}$ and $\bigwedge V = V$. Thus, an element $v \in \bigwedge V$ is a sequence $v=(v\_{0},v\_{1},...)$, with $v\_{n}\in \bigwedge^{n}V$, with all but finitely many nonzero entries.. $\bigwedge^{n}V$ can be realized as the subspace of all skew-symmetric tensors of $\overbrace{V\otimes \cdots \otimes V}^{\text{n times}}$.


**My question is:** If $V$ is replaced by a **normed** vector space $U$, is $\bigwedge^{n}U$ defined in the same algebraic way as before? I know that I can induce a norm on $\bigwedge^{n}U$ from $U$ but I realy don't know anything about the construction of exterior powers of normed vector spaces. Is that any different from the algebraic one? Also, if $U$ is Banach, is $\bigwedge U$ also Banach or we need to complete it?



Exterior algebra of normed spaces
If $V$ is a Hilbert space there is a standard notion of alternating tensor. First, we have a definition of full tensor products of Hilbert spaces such that if $\{e\_i\}$ is an orthonormal basis of $V$ then $\{e\_i \otimes e\_j\}$ is an orthonormal basis of $V \otimes V$ (and similarly for more than two factors). Then we have a notion of symmetric and antisymmetric parts of $V^{\otimes n}$ coming from the natural action of $S\_n$ on this space. The antisymmetric part is considered as a space of alternating tensor over $V$, and the exterior algebra is taken to be the direct sum of the alternating tensor powers of $V$. This is the antisymmetric or "bosonic" [Fock space](https://en.m.wikipedia.org/wiki/Fock_space).


I don't know about other Banach spaces. Anthony Quas gives a reference for this in the comments. But in general there are many reasonable ways to norm tensor products of Banach spaces, so I don't think it's fair to expect there to be a really canonical answer.


