Wedge products and exterior powers are discussed in W. Greub's book Multilinear algebra as follows.

**Definition:** Let $E$ be an arbitrary vector space and $p \ge 2$. Then a vector space $\bigwedge^{p}E$ together with a skew-symmetric $p$-linear map $\bigwedge^{p}: E\times \cdots \times E \to \bigwedge^{p}E$ is called a $p$-th exterior power of $E$ if the following conditions are satisfied:

(1) The vectors $\bigwedge^{p}(x_{1},\dotsc,x_{p})\mathrel{:=} x_{1}\wedge \dotsb \wedge x_{p}$ generate $\bigwedge^{p}E$.

(2) If $\psi$ is any skew-symmetric $p$ linear mapping of $\overbrace{E\times \dotsb \times E}^{\text{$p$ times}}$ into an arbitrary vector space $F$, then there exists a linear map $f\colon \bigwedge^{p}E \to F$ such that $\psi = f\circ \bigwedge^{p}$.

Now, we set: \begin{equation} \bigwedge E \mathrel{:=} \bigoplus_{n=0}^{\infty} \bigwedge^{p}E, \tag{1}\label{1} \end{equation} where $\bigwedge^{0}E \mathrel{:=} \mathbb{C}$ and $\bigwedge^{1}E \mathrel{:=} E$.

Identifying each $\bigwedge^{p}E$ with its image under the canonical injection $i_{p}\colon\bigwedge^{p}E \to \bigwedge E$, we can write $\bigwedge E = \sum_{p=0}^{\infty}\bigwedge^{p}E$. In other words, elements of $\bigwedge E$ can be thought as sequences $(v_{0},v_{1},\dotsc)$ where $v_{p} \in \bigwedge^{p}E$ for each $p\in \mathbb{N}$. Furthermore, there is a uniquely determined multiplication on $\bigwedge E$ such that the following rules hold: \begin{gather*} (x_{1}\wedge \cdots \wedge x_{p})(x_{p+1}\wedge \dotsb \wedge x_{p+q}) = x_{1}\wedge \cdots \wedge x_{p+q} \\ 1(x_{1}\wedge \cdots \wedge x_{p}) = (x_{1}\wedge \dotsb \wedge x_{p})1 = x_{1}\wedge \cdots \wedge x_{p}. \end{gather*} This turns $\bigwedge E$ into an algebra, which is called exterior (or Grassmann) algebra.

Note that Greub's construction considers arbitrary vector spaces, so that, in particular, we can take $E$ to be infinite dimensional.

Grassmann algebras are used by physicists to study fermionic systems. While searching for some material on Grassmann algebras of infinite dimensional vector spaces, I found lecture notes Fermionic functional integrals and the renormalization group by Feldman, Knörrer and Trubowitz, which has an appendix (page 75) on this topic. Their construction seems interesting, but I'm having trouble trying to relate it with Greub's construction.

The first part of their notes discusses Grassmann algebras of finite dimensional vector spaces. Then, the cited appendix starts with the statement that in order to further generalize it to infinite dimensional vector spaces we need to add a topology on these spaces. This seem not to be necessary in the general case, since Greub's construction does not consider topological vector spaces. However, I think they might have physical motivations in which the addition of a topology might be important. Their construction is as follows.

Let $I$ be a countable set. The Grassmann algebra will be generated by vector from the vector space: $$E\mathrel{:=} \ell^{1}(I)\mathrel{:=}\{\alpha\colon I \to \mathbb{C}\mathrel: \sum_{i\in I}\lvert a_{i}\rvert < +\infty\}.$$ $E$ is a Banach space with norm $\|\alpha\| \mathrel{:=}\sum_{i\in I}\lvert a_{i}\rvert$. Let $\mathcal{J}$ be the set of all finite subsets of $I$, including the empty set. Take $$\mathcal{U}(I) = \ell^{1}(\mathcal{J}) \mathrel{:=}\{\alpha\colon \mathcal{J} \to \mathbb{C}\mathrel: \sum_{I\in \mathcal{J}}\lvert a_{I}\rvert<+\infty\}$$ where $a_{I} \mathrel{:=} a_{i_{1}}\dotsb a_{i_{p}}$, $I=\{i_{1},...,i_{p}\}$. Then $\mathcal{U}(I)$ is a Banach space with norm $\|\alpha\| = \sum_{I\in \mathcal{J}}\lvert a_{I}\rvert$ and, when equipped with the product: $$(\alpha \beta)_{I} \mathrel{:=}\sum_{J\subset I} \operatorname{sign}(J, I\setminus J)\alpha_{J}\beta_{I\setminus J},$$ it becomes an algebra which is called the Grassmann algebra.

With all this being said, let me get to the questions.

Feldman, Knörrer and Trubowitz's construction might not be the most general construction there is (I don't know actually, but I think it's not as I justified before). However, I'd expect their construction to be at least a particular case of Greub's general construction. However, I don't seem to be able to relate these two since the definition of $\mathcal{U}(I)$ strongly depends on its topology. So **is the second construction a particular case of the first one?** **If not, why not?** **Does it have to do with the hypothesis of $E$ to be a topological vector space? Does the topology on $E$ change the definitions of objects used on Greub's construction?**

**NOTE:** When I ask "does the topology on $E$ changes the definitions of objects on Greub's constructions?", I mean the following. If $E$ is a vector space, $\bigoplus_{n=0}^{\infty}E$ is the space of all sequences $x=(x_{0},x_{1},\dotsc)$, $x_{i} \in E$, with all but finitely many nonzero entries. If $E= \mathcal{H}$ is a Hibert space, on the other hand, $\bigoplus_{n=0}^{\infty}\mathcal{H}$ is the space of sequences with $\|x\|^{2}:=\sum_{n=0}^{\infty}\| x_{i}\|^{2}_{\mathcal{H}}<+\infty$. Thus, although $\mathcal{H}$ is itself a vector space, the norm on $\mathcal{H}$ allows us to define the direct sum in alternative way. In other words, the topology on $\mathcal{H}$ makes the difference when we define direct sums. Maybe the use of Banach spaces by Feldman, Knörrer and Trubowitz implies some modifications like this, say, to define the direct sum (\ref{1}) in an alternative way, so these two constructions might be isomorphic or something like this.

**ADDED:** Does anyone know this particular construction from Feldman, Trubowitz and Knörrer? Any references on this approach would be really appreciated!

is not the same objectas the "infinite sum of Hilbert spaces". The point is that the usual direct sum construction, when applied to a sequence of complete normed spaces, usually does not yield a complete normed space. So if you want your algebraic constructions to start with complete things and produce complete things, they need to be modified $\endgroup$`\wedge`

is the binary operation; you want $\bigwedge$`\bigwedge`

for the unary prefix operation, like $\bigwedge E$. Also,`\mbox`

doesn't size well in superscripts, whereas`\text`

does: compare $\overbrace{E \times \dotsb \times E}^{\mbox{$p$ times}}$ (with`\mbox`

) to $\overbrace{E \times \dotsb \times E}^{\text{$p$ times}}$ (with`\text`

). I have edited accordingly. $\endgroup$