This is the Sequel of a preceeding Question of mine
Exterior product is a ring, showing that the multiplication has desired qualities
I have explained the Notation of the Symbols I am using in this Question.
We define
$$\bigwedge V=\bigoplus_{r\in\mathbb{N}}\bigwedge^{r}V:=\{(\lambda_r)_{r\in\mathbb{N}}\in\prod_{r\in\mathbb{N}}\bigwedge^{r}V:\lambda_r=0\text{ for allmost all r}\}$$
Then one can define a multiplication on the vectorspace $\bigwedge V$ with the help of the unique biliear map $\wedge$ that has the properties:
$$\wedge:\bigwedge^{r}V\times\bigwedge^{s}V\rightarrow\bigwedge^{r+s}V$$
$$\wedge((a_1\wedge...\wedge a_r),(b_1\wedge....\wedge b_s))= a_1\wedge ...\wedge a_r\wedge b_1\wedge ...\wedge b_s$$
(see for more Details of this in the linked question)
making the set $\bigwedge V$ a ring
The books only goes this far and does not actualy define the desired multiplication $$\cdot:\bigwedge V \times \bigwedge V \rightarrow \bigwedge V$$
such that $(a\cdot b)\cdot c=a\cdot(b\cdot c)$ and $(a+b)\cdot c= a\cdot c + b\cdot c$ and $a\cdot(b+c)=a\cdot b + a\cdot c$
I suspect that the desired multiplication using the helping function $\wedge$ is
given with
$$\cdot(a,b)=\cdot((\lambda_r),(\lambda'_r)):=\sum_{s\in \{r\in\mathbb{N}:\lambda_r\neq 0\}}\sum_{s'\in \{r\in\mathbb{N}:\lambda'_r\neq 0\}}\lambda_s\wedge\lambda'_{s'}$$
But the proof seems very technical and I would like to read it. Can someone give me a source? My book does not contain it and I don't know how to search this specific proof online by myslef.
Yes, your formula gives the product on $\bigwedge V$. However, one should not be afraid of it. It is very similar to multiplication of polynomials: if $R$ is a ring, then $R[x] = \bigoplus_{n=0}^\infty R \cdot x^n$ as a module over $R$, with product induced by $$ R\cdot x^r \times R \cdot x^s \to R \cdot x^{r+s}$$ given by $ ax^r \times b x^s \mapsto (ab)x^{r+s}$. The multiplication on $R[x]$ is then defined as the product termwise: to multiply two polynomials, distribute and multiply monomial-by-monomial in each degree.
Standard properties of the multiplication on $\bigwedge V$ follow from the corresponding properties for the products $\bigwedge^r V \times \bigwedge^s V \to \bigwedge^{r+s} V$. For instance, distributivity follows since $(( \lambda_r) + (\lambda'_r))_s = \lambda_s + \lambda'_s$ and then from distributivity of wedge.