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I just showed, using the Baire category theorem, that if $V$ is infinite-dimensional Banach space, then its Hamel basis has to be uncountable. The next question was:

Is there a norm for the space of real-valued polynomials $\mathbb{R}[X]$ s.t. $(\mathbb{R}[X], \lVert\cdot\rVert)$ is Banach space?

My answer is no. $\mathbb{R}[X]$ is infinite-dimensional and it has a Hamel basis $\{x^n:n\in \mathbb{N}\}$, which is countable. Therefore $(\mathbb{R}[X], \lVert\cdot\rVert)$ cannot be complete w.r.t. any norm.

Is this enough? I mean, does the result I proved earlier hold for any Hamel basis of $V$? If I can find one countable Hamel basis for $V$, then, if $V$ is infinite dimensional, it cannot be complete?

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    Yes, the cardinality of a basis (in the algebraic sense) is well-defined, all bases of a vector space have the same cardinality. – 2017-02-15
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    The proof based on Baire works for *every* Hamel basis, your argumentation is correct. – 2017-02-15
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    Is it a trivial fact that every basis has the same cardinality? – 2017-02-15
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    @Infinitebig, yes -- basic linear algebra. Take an isomorphism. What does it send a basis to? – 2017-02-15
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    Isomorphism maps basis to basis. But if I have two Hamel basis $A$ and $B$, does it necessary mean that I have isomorphism between them? – 2017-02-15

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