Why is the dual space of $\mathbb{R}$ a subset of the dual space of $\mathbb{Q}$?

For the dual space I am looking at every linear functional of the form $$L:X\rightarrow\mathbb{Q}$$ for $X$ either $\mathbb{R}$ or $\mathbb{Q}$.

closed as unclear what you're asking by Andrés E. Caicedo, Gibbs, Vladhagen, amWhy, Chris Custer Oct 10 at 16:10

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  • 3
    What is “dual space” in this context? – José Carlos Santos Oct 10 at 13:53
  • consisting of all linear functionals on $\mathbb{R}$, together with the vector space structure of pointwise addition and scalar multiplication by constants – geremia Oct 10 at 13:59
  • Linear functionnals over which field? – José Carlos Santos Oct 10 at 14:01
  • Clarified in the question now – geremia Oct 10 at 14:13
  • Since the irrational numbers do not form a vector space (over $\mathbb R$ or $\mathbb Q$), I don't understand your question. – José Carlos Santos Oct 10 at 14:15