natural bilinear extension

Question

The outer Product between Forms is a map

$\wedge: \bigwedge^r V^* \times \bigwedge^s V^* \rightarrow \bigwedge^{r+s} V^*, (\alpha, \beta) \mapsto \alpha \wedge \beta$

My Analysis III book Amann says that the outer Product is naturally extendable to the direct sum

$\bigwedge V^*:= \bigoplus_{k=0}^\infty \bigwedge ^k V^* = \{ (x_k)_{k \geq 0} | x_k \in \bigwedge ^k V^* \}$

What is meant by this natural extension?

Answer 1

If I understand you and the book, it means that outer (or exterior) power of $V^{*}$ could be extended as an algebra to the exterior algebra $\bigwedge V^{*}$. Exterior power $\bigwedge^k V^* $ itself is a vector space, but since there is natural map (as you've written), all exterior powers together could form an exterior algebra $\bigwedge V^*$. Product of the exterior algebra is defined as that map above.