Yes, I see it now (the conclusions concerning $\mathbb{R}^n$ follow in a more general form from the Mazur-Ulam theorem, though this is a bit overkilling here). Thanks! P.S.: you may want to edit a minor error in your LaTeX (I think a symbol = is missing where you define the $d_i$'s).

@Anatoly Kochubei. Thanks for your answer and the references. Indeed, the 2nd paper is interesting to me, though it's still, say, too "localised" with respect to the kind of stuff that I've in mind. In any case, I will read it with more attention. Also, let me remark once more that I've proved so far only basic results (essentially the ones listed in the OP and a few minor others). For one thing, I did not even have a way to show that, under reasonable assumptions, this "generalized spectrum" is non-empty.

(...) your framework somehow more flexible (starting with your definitions). That's that. Also, I should have probably remarked that my reflection about completeness is not the true point here, though it served as a trigger (and it is indeed a feature of the kind of generalisation that I've in mind). One thing for sure, I'm going to edit the OP to second your suggestion, highlight my requests, and (try to) clean up some passages. As for the rest, could you elaborate your last comment above? Thanks again.

@Yemon Choi. Thanks for your comments. I see your point, and I've nothing to object. Yet, my view is different, or - better - complementary. I'm mainly interested in approximation theory, where the (still naïve) basic idea is, say, that you're not really interested in solving your equations in an "exact way", but you content to do it within the bounds of an arbitrary $\varepsilon > 0$. So, in this setting (as well as in others), you may want to rule out any unnecessary complications arising from the fact that favorable properties are simply lacking (as a "fact of life"), and hence make (...)

As a consequence, you're decidedly right with your last remark too, and the original question is no longer that interesting (at least, as is). Also, your guess is right - I was implicitly assuming the existence of an (extended) Schauder basis. Now, what's the right way to go in this case? Should I edit the OP and update it accordingly? Or would it be better to open a new topic?

@Pietro Majer. Sorry for the big delay in replying! I was indeed sloppy in many respects, let's take this step-by-step. First, the question about the "realisation" of a complex normed space (btw, is there a standard naming for referring to this process? We have the term complexification for the dual one, but I'm not aware of any analogue). Finally, I agree that it's enough to deal with the real case (for gloomy reasons, I thought that some extra compatibility between the original norm and the one obtained by restriction of the scalar field should have been necessary for this to work).

For the sake of curiosity, aren't all of your isomorphisms already surjective by definition? :)

So nice, I would say! Thank you, BJ, this is very useful; though I had hoped for a positive answer, you just let me realise something that I had completely overlooked. I know, I know, I shouldn't fall in love with an idea, and continue forgetting the golden rule: when you can't prove something, break it down and try to seek a counterexample - in the worst case, you will get further insight on your problem. Thanks again!

I'm especially looking at the existence of a weakly convergence subsequence since it seems to me the most natural way to go - as the conditions in the problem are stated in terms of the functionals in the (continuous) dual space. Does this (make more sense and) somehow clarify the question?

@BJ. In fact, I'd like to prove that a certain Cauchy sequence (in a possibly incomplete normed space) is norm convergent. I just know that the sequence satisfies a few conditions (as given in the OP). I'm wondering if all of them together imply some well-known (but still not so well-known to me) sufficient condition for the existence of a weakly convergent subsequence, indeed whether or not all of them are sufficient to establish the existence of a weakly convergent subsequence (which would be enough). Also, I see by now that I phrased the question in the worst of all possible ways...