I am reading Elements of Algebra by Bourbaki and I'm having difficulty understanding this passage (page $63$, chapter $10$, section $4$):
Let $M$ be a vector space over $N$; for each $\sigma\in\Gamma$ let $M^\sigma$ be the vector space over $N$ with the same underlying additive structure group as $M$, with the external law $(\lambda,x)\mapsto\sigma(\lambda)x$. Write $V=\prod_{\sigma\in\Gamma}$; the underlying additive group of $V$ is that of all mappings of $\Gamma$ into $M$, with the external law defined by $$(\lambda.h)(\sigma)=\sigma(\lambda)h(\sigma)~~~(\lambda\in N,h\in V,\sigma\in\Gamma)\tag5$$ (The product $\sigma(\lambda)h(\sigma)$ is calculated in the vector space $M$.) Further, we define on $N\otimes_K M$ a vector space structure over $N$ by the formula $$\lambda\left(\sum_i\mu_i\otimes x_i\right)=\sum_i\lambda\mu_i\otimes x_i$$ Finally we denote by $\psi$ the $K$-linear mapping of $N\otimes_k M$ into $V$ characterized by the relation $$\psi(\lambda\otimes x)(\sigma)=\sigma(\lambda).x\tag6$$ for $\lambda\in N$, $x\in M$ and $\sigma\in\Gamma$. It is clear that $\psi$ is $N$-linear
Note that $N$ is a Galois extension over the field $K$, and $\Gamma$ is the corresponding Galois group.
I don't understand how they define the additive structure on $V$. Also, I don't understand what $h(\sigma)$ is. How is $h\in V$ acting on $\sigma \in \Gamma$? Any help deciphering this would be grand.
The elements of $V$ are maps $h:\Gamma\to M$. If $h$ and $h'$ are such maps, then $h+h'$ is defined by $$(h+h')(\sigma)=h(\sigma)+h'(\sigma)$$ (pointwise addition). I would say $n\in N$ acts on $h\in V$ (via the given formula (5)) rather than $h\in V$ acts on $\sigma\in\Gamma$.