# Why is R* “smaller” than Q* [closed]

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 CusterOct 10 '18 at 16:10

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• What is “dual space” in this context? – José Carlos Santos Oct 10 '18 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 '18 at 13:59
• Linear functionnals over which field? – José Carlos Santos Oct 10 '18 at 14:01
• Clarified in the question now – geremia Oct 10 '18 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 '18 at 14:15