$\mathbb{R}$ is a vector space over itself, over the embedded copy of $\mathbb{Q}$ by restriction of scalars, and over $\mathbb{Q}$ itself via the embedding

Let $\mathbb{R}$ be the real numbers (The real numbers), a field (The reals form a field) and an ordered field (The reals form a totally ordered field, Ordered field), and let $\mathbb{Q}$ be the rationals, a field (The rationals form a field).

1. $\mathbb{R}$ is a vector space over itself (Vector space over a field): the vectors are the reals, the vector addition is the field addition, the zero vector is $0$, and the scalar multiplication is the field multiplication.
2. Let $\iota : \mathbb{Q} \to \mathbb{R}$ be the unique field homomorphism (The unique embedding of ℚ into an ordered field, Field homomorphism and embedding), which is injective and order-preserving. Its image $\mathbb{Q}_{\mathbb{R}} := \iota[\mathbb{Q}] = \{\, \iota(q) : q \in \mathbb{Q} \,\}$ is a subfield of $\mathbb{R}$ (Subfield: a subring of a field closed under inverses of its nonzero elements, and therefore a field with the restricted operations), and by restriction of scalars (A field is a vector space over itself, and over any subfield $K \subseteq F$ every $F$-vector space is a $K$-vector space by restricting the scalars) $\mathbb{R}$ is a vector space over $\mathbb{Q}_{\mathbb{R}}$, the scalar multiplication being the field multiplication restricted to $\mathbb{Q}_{\mathbb{R}} \times \mathbb{R}$.
3. Setting $q \cdot x := \iota(q)\,x$ for $q \in \mathbb{Q}$ and $x \in \mathbb{R}$ makes $\mathbb{R}$ a vector space over $\mathbb{Q}$ itself.

The subfield is the image, not $\mathbb{Q}$. $\mathbb{Q}$ is not a subset of $\mathbb{R}$ in this library: a rational is a class of pairs of integers and a real is a class of Cauchy sequences of rationals (The real numbers). What sits inside $\mathbb{R}$ as a subfield is the image $\mathbb{Q}_{\mathbb{R}}$ of the embedding, and claim 2 is a statement about that image. Claim 3 is the statement about $\mathbb{Q}$ itself, and it is proved directly rather than by restricting scalars, because restriction of scalars requires a subfield.

• Three structures on one set. The vectors are the same reals throughout and the addition is the same in all three cases; what changes is which scalars are allowed to act. Claims 2 and 3 differ only in bookkeeping: the scalars are the elements of $\mathbb{Q}_{\mathbb{R}}$ in one and the elements of $\mathbb{Q}$ in the other, and $\iota$ matches them up bijectively, being injective onto its image.

• Nothing here is said about size. How big $\mathbb{R}$ is as a vector space over $\mathbb{Q}$ is a question about bases and dimension, which are developed on a later page; no claim about either is made above, and the verification uses neither.

• Why the ordered field hypothesis appears at all. The embedding $\iota$ is supplied by The unique embedding of ℚ into an ordered field, which is stated for an ordered field, and $\mathbb{R}$ is one. The order plays no further role: once $\iota$ is in hand, every step above uses only that it is a field homomorphism.