# What is so special about the real and complex numbers?

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?

• 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?
– Moya
Commented Mar 14, 2015 at 4:29
• Having every property (and perhaps more!) that reals (or complex numbers) do, such as completeness, denseness, totally orderedness, infiniteness, etc. Commented Mar 14, 2015 at 4:31
• 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.
– Moya
Commented Mar 14, 2015 at 4:33
• 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.
– KCd
Commented Mar 14, 2015 at 4:54
• This seems like three rather separate questions, so it should be asked as three separate questions. Commented Mar 14, 2015 at 11:09

• 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. Commented Mar 14, 2015 at 4:52