Theorem of Hamel/Examples/Exercise

What does the Theorem of Hamel mean in the following examples?

1. The real numbers ${}\mathbb {R}$ as ${}\mathbb {Q}$-vector space.
2. The set of all real numbers

   ${}\mathbb {R} ^{\mathbb {N} }={\left\{{\left(x_{n}\right)}_{n\in \mathbb {N} }\mid x_{n}\in \mathbb {R} \right\}}\,.$

3. The set of all continuous functions from ${}\mathbb {R}$ to ${}\mathbb {R}$.

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