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Give me some examples of basis for $\mathbb R$ (as vector space over field $\mathbb F=\mathbb Q$).

Thanks.

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    Please give me orders; I really enjoy being told what to do. Oh, wait. I *don't*. Never mind...2012-02-24

1 Answers 1

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While it is perfectly reasonable to write down a basis for a finitely dimensional vector space, it is not always possible to write one for infinitely dimensional vector spaces.

In fact the assertion that every vector space has a basis is equivalent to the axiom of choice. This does not mean that every infinitely dimensional space has no basis. For example $\mathbb R[x]$ as a vector space over $\mathbb R$ is infinitely dimensional, but it has a basis - $\{x^n\mid n\in\mathbb N\}$.

There are models of set theory without the axiom of choice in which there is no basis for $\mathbb R$ over $\mathbb Q$, which means that one cannot just "write down" such basis, but rather that one can prove the existence of a basis in a non-constructive manner such as Zorn's lemma.

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    BTW, I think your second paragraph can use some clarifying. I think you mean to say: "This does not mean that a basis can never be explicitly written down for an infinite dimensional vector space." And perhaps the example of $\mathbb{R}[x]$ can be given too.2012-02-24