The space of positive real numbers

Task number: 2504

Let \(\mathbb R^+\) be all the positive real numbers and define operation \(\oplus\) on \(\mathbb R^+\) and \(\odot: \mathbb Q\times \mathbb R^+\to \mathbb R^+\) as follows: \[ u\oplus v = uv, \hspace{2cm} a\odot u = u^a \] Is it true, that \((\mathbb R^+,\oplus,\odot)\) is a vector space over the rational numbers \(\mathbb Q\)?

  • Resolution

    It is true – it suffices to verify all the axioms of the vector spaces.

    Let \(\mathbf 0=1\) be the neutral element and (SO) proceed as follow \( u\oplus \mathbf 0= u \oplus 1 = u\cdot 1=u\).

    For the axiom (SI) lets choose the oposite vector as \(\ominus u=\frac1u\in \mathbb R^+\) and then we proceed with
    \(u\oplus (\ominus u)=u\cdot \frac1u=1=\mathbf 0\).

Difficulty level: Easy task (using definitions and simple reasoning)
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