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When I was studying linear algebra, the first thing we were introduced was the idea of fields. In studying analysis (and when studying inner product spaces etc), we restricted our possible fields to that of reals or complexes. So... here is my question.

  1. In defining the inner product space, one restricts the underlying field to either real or complex. It seems to me that in defining the inner product space, it was only necessary that the field be totally ordered. Ah, I forgot, there is this conjugate symmetric property. So, how do we characterize the field so that "conjugacy" makes sense? (involution perhaps?)

  2. Okay, how about the normed vector space? I guess, from the name normed vector space, one can define the norm on any totally ordered field? How about the Banach space? By adding the completeness (analytic) condition, can we say that the Banach space can be defined on any totally ordered, complete field?

  3. Lastly, as the title suggests, what is so special about the real and complex numbers? Okay, I do know certain properties of these fields, viz. reals are complete, totally ordered, dense, infinite in both directions, etc. But do there exist fields that are NOT isomorphic to these fields (real and complex) such that they behave EXACTLY the same as the real and complex?

Thanks in advance!

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  • $\begingroup$ For your last question, what do you mean behave exactly the same?For the real numbers, it is the unique totally ordered complete Archimedean field, so would you be asking which of these we could relax so that linear algebra/normed spaces, work the same? $\endgroup$
    – Moya
    Commented Mar 14, 2015 at 4:29
  • $\begingroup$ Having every property (and perhaps more!) that reals (or complex numbers) do, such as completeness, denseness, totally orderedness, infiniteness, etc. $\endgroup$
    – insignia
    Commented Mar 14, 2015 at 4:31
  • $\begingroup$ Sorry, accidentally pressed enter too soon, so there's more to my comment. But I can add: Any field that has the reals as a subfield and has square root of $1$ has a field which is isomorphic to $\mathbb{C}$, so again this is essentially also uniquely defined. $\endgroup$
    – Moya
    Commented Mar 14, 2015 at 4:33
  • $\begingroup$ There are $p$-adic Banach spaces, where the scalars are in a $p$-adic field (complete extension field of $\mathbf Q_p$), but not really $p$-adic Hilbert spaces. $\endgroup$
    – KCd
    Commented Mar 14, 2015 at 4:54
  • $\begingroup$ This seems like three rather separate questions, so it should be asked as three separate questions. $\endgroup$ Commented Mar 14, 2015 at 11:09

2 Answers 2

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In characteristic zero Q is the smallest field (in the sense that this is contained as a subfield in all of them). And when we complete it (analytically) we get the real numbers and when we take the algebraic closure of the latter we get C. In this sense real numbers and complex numbers are the smallest field to do analysis.

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  • $\begingroup$ Pretty much what I was getting at in the comments. Thanks! $\endgroup$
    – Moya
    Commented Mar 14, 2015 at 4:34
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    $\begingroup$ Just saying that when you complete $\mathbb{Q}$ you get $\mathbb{R}$ is not sufficient, because there are other metrics on $\mathbb{Q}$ with other completions, namely the $p$-adic completions. $\endgroup$
    – Lee Mosher
    Commented Mar 14, 2015 at 4:52
  • $\begingroup$ Thanks Mosher, for pointing out the lack of precision. $\endgroup$ Commented Mar 14, 2015 at 8:43
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A bit of a wider view: if one wants to work over a field of characteristic zero, one is forced to start with the rationals, as already mentioned. To do analysis one needs a metric that is coupled with the field operations. The usual way to achieve this is by taking an absolute value function on the field. For the rationals all absolute value functions are known: they are the ordinary absolute value function and for each prime number the p-adic value function. Now for each of them one can take the completion, thus getting the reals and the p-adic numbers for each p. Next it is desirable to have algebraically closed fields to do function theory for example. In the case of the reals one gets the complex numbers. In the case of the p-adics however the algebraic closure is not complete anymore. Therefore to get a field, in which one can do analysis, one has to take completion again, which then yields a complete algebraically closed field comparable to the complex numbers for each prime p.

After all one can (try to) do analysis either over the reals or over the p-adics, one can (try to) do function theory over the complex numbers or over the fields obtained by the two-step-procedure described above. In particular there are generalizations of for example funtional analysis to the p-adic case. The subject however is quite problematic due to their topological properties, being totally disconnected at the first place to mention here.

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