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, Namaste, Chris Custer Oct 10 '18 at 16:10

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    $\begingroup$ What is “dual space” in this context? $\endgroup$ – José Carlos Santos Oct 10 '18 at 13:53
  • $\begingroup$ consisting of all linear functionals on $\mathbb{R}$, together with the vector space structure of pointwise addition and scalar multiplication by constants $\endgroup$ – geremia Oct 10 '18 at 13:59
  • $\begingroup$ Linear functionnals over which field? $\endgroup$ – José Carlos Santos Oct 10 '18 at 14:01
  • $\begingroup$ Clarified in the question now $\endgroup$ – geremia Oct 10 '18 at 14:13
  • $\begingroup$ Since the irrational numbers do not form a vector space (over $\mathbb R$ or $\mathbb Q$), I don't understand your question. $\endgroup$ – José Carlos Santos Oct 10 '18 at 14:15