I am writing something about vector spaces, but I am not sure if I am using the appropriate notation.
Suppose we have a vector space $V$ over $\mathbb{Q}$ generated by formal symbols $x_1,..,x_d$. I would write that the basis of $V$ is $B=\{x_1,..,x_d\}$ and that $V$ itself is $\langle x_1,..,x_d\rangle_\mathbb{Q}$. Also I write for example, $V \otimes \mathbb{R}=\langle x_1,..,x_d \rangle_\mathbb{R}$.
And suppose we have a subspace of $V$, generated by some elements from the basis, I would write $W=\langle v_1,..,v_k \rangle_\mathbb{Q}$ such that for all $1 \leq j \leq k$, $v_i=x_j$ for some $1 \leq j \leq d$. Or should I write $v_i \in B$ for all $i$?
Is there a nicer/more conventional notation?
Usually one says that $V$ is a vector space over $K$, with $K$-basis $(x_1,\ldots ,x_d)$. Then one can specify the field, i.e., consider $V$ as a $\mathbb{Q}$-vector space, or as $\mathbb{R}$-vector space. Some people write $$ V_{\mathbb{R}}=V_{\mathbb{Q}}\otimes_{\mathbb{R}}\mathbb{R}. $$