Suppose we have an induced representation of $\theta: H \to GL(W)$, we define the space $$ V := \mathbb{C}[G] \otimes_{\mathbb{C}[H]} W, $$ $V$ has as a basis $\{e_r \otimes_{\mathbb{C}[H]} w\}_{r \in \mathcal{R}, w \in B}$ with $\mathcal{R}$ a complete set of representatives of $G$ and $B$ a basis for $W$. The representations is defined as $g \cdot (e_r \otimes_{\mathbb{C}[H]} w) = (e_{gr} \otimes_{\mathbb{C}[H]} w)$.
Then, in order to compute the character, we define the subspaces: $$ V_g = e_g \otimes_{\mathbb{C}[H]} W \subseteq V $$ (notice that this only depends on the coset).
My question is: how does it follow that this is a subspace? In order to show it, we need to check it is closed under addition and scalar multiplication, for example is $$ e_g \otimes_{\mathbb{C}[H]} w + e_g \otimes_{\mathbb{C}[H]} w' \in V_g? $$ And scalar multiplication as well. I realized that I don't even know how these two (scalar multiplication and addition) are defined in this highly complex $V$ (I don't even know over which field we should use scalar multiplication @_@, $\mathbb{C}$? $\mathbb{C}[H]$?), since everything is constructed so abstractly and indirectly. Therefore I have no idea how to check that $V_g$ is indeed closed under addition and scalar multiplication. Can someone enlight me on this case?