If I understand the whole Hamel basis idea correctly, there exists one such basis $B = \{v_\alpha\}_{\alpha \in I}$ for ${\mathbb R}$ (herein construed as a vector space of ${\mathbb Q}$), such that $1 = v_0 \in B$. (Here I'm using $I$ to denote some suitable index set; the $0$ subscript in $v_0$ just stands for one element of $I$.)
The span of $v_0 = 1$ in ${\mathbb R}$ is thus ${\mathbb Q}$.
This means that every $x \in {\mathbb R}$ can be expressed uniquely as a linear combination of the form:
$$x = q_0 + \sum_{\alpha \in I \backslash \{0\}} q_{\alpha} \cdot v_{\alpha}$$
Let $P_{\mathbb Q}:{\mathbb R}\rightarrow {\mathbb Q}$ be the projection of ${\mathbb R}$ onto ${\mathbb Q}$. Namely, using the same notation as in the expression for the decomposition of any real $x$ above,
$$P_{\mathbb Q}(x) = q_0$$
My first question is:
can it be proved or disproved whether the projection $P_{\mathbb Q}$, or at least its nullspace, ${\mathcal N}(P_{\mathbb Q})$, is independent of the choice of $B\backslash \{1\}$?
My second question is:
assuming that ${\mathcal N}(P_{\mathbb Q})$ is in fact independent of the choice of $B\backslash \{1\}$, is there a special name given to this subspace of ${\mathbb R}$?
