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@Pietro Majer. I'm not yet convinced with your 1st comment on the finite dimensional case. Let $\mathbf{X} \equiv (X, \|\cdot\|)$ be a complex normed space. Of course, we agree that $X$ can be as well regarded as a real vector space, and indeed $\dim_\mathbb{R}(X) = 2\, \dim_\mathbb{C}(X)$. But I'm not so sure that $\mathbf{X}$ can also be seen as a real normed space (essentially because $|a|+|b| \ne |a+ib|$ for arbitrary $a,b\in\mathbb{R}$). What do I miss here? Still, as you pointed out below, the finite dimensional case is settled by looking at the Hausdorff-Besicovitch dimension.
Note. By error, I deleted my previous comment (dated 25 Aug 2011). Now, I've tried to remember what it should be and posted it again. Sorry for the inconvenience! @Samuele. Not really what I was seeking, but still useful. Thanks!
@Pietro. You're absolutely right with your 2nd comment! The key point is that the Hamel dimension of an $\infty$-dimensional (real or complex) Banach space cannot be less than $|\mathbb{R}|$ (even if the CH fails!) as proved in H. E. Lacey, The Hamel Dimension of any Infinite Dimensional Separable Banach Space is $\mathfrak{c}$, The AMM, Vol. 80 (1973), p. 298. I confess, this is somehow surprising for me as I really thought the answer should not depend on the completeness of the space, and I'm editing the OP accordingly to pose the question in the right (normed) setting.
Ops! I just missed the "II" in the title... This is supposed to explain why I couldn't find a lot of the things that you had mentioned in your (second) answer. :)
@Bill Johnson. Thank you for your contribution and the reference. Just for the record, a further possibility, in the finite-dimensional case, is provided by the so-called weak Banach-Mazur distance as given in N. Tomczak-Jaegermann, "The weak distance between Banach spaces", Math. Nachr., 119 (1984), pp. 291-307.
Ah! I had completely missed that... The only difference between your proof and mine is that I'm stating it in the language of $\varepsilon$-pseudospectra, but it makes no real difference, indeed.
