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I've been thinking about this, I don't know how to look up anything similar, so here I am asking a question. Specifically, is there any space $X$ with the following properties:

  • Algebraic structure: The first thing we demand is that $X$ is a commutative ring. Later on, I'll explain why the field structure is also demanded.
  • Topology. Obviously we need a notion of closeness on our space $X$. What is more is that we want to correlate the distance of points $a,b$ to the element $a-b$ of $X$. That being said, we demand that there exists something like a norm on $X$ (note: $X$ is not necessarily a vector space over $\mathbb{R}$), namely $\|\cdot\|$, that has the following properties:

(a) $\|x\|=0$ iff $x=0$.

(b) $\|-x\|=\|x\|$

(c) $\|x+y\|\leq\|x\|+\|y\|$

So $\|\cdot\|$ actually induces a metric on $X$: set $d(x,y):=\|x-y\|$ .

Short proof: 1) $d(x,y)=0$ iff $\|x-y\|=0$ iff $x-y=0$ iff $x=y$. 2) $d(x,y)=\|x-y\|=\|y-x\|=d(y,x)$. 3) $d(x,y)=\|x-y\|=\|x-z+z-y\|\leq\|x-z\|+\|z-y\|=d(x,z)+d(z,y)$.

  • We demand the ability to differentiate some functions $f:X\to X$, so we want a field structure, since we have to deal with limits of the form $\displaystyle{\frac{f(x_n)-f(x)}{x_n-x}}$.

I am not adding anything extra for integration, since measures can be attached to any set. Before you comment or answer about the Frobenius theorem, note that there is no need for $X$ to be a vector space over $\mathbb{R}$ ,or, even if it was, no need of being of finite dimension over $\mathbb{R}$. I also know some basics about differentiable manifolds, so any info on this is not necessary. I'm looking for an exact answer on the question, and, differentiable manifolds, although close (in the sense that there can be tons of analysis performed on them), are far from having those properties.

Part 2: (refering to the members of the community with experience on research)

If there are known examples of such spaces, are they interesting at all?

P.S. I couldn't think of any reason this question is trivial or silly, but if it is, feel free to say so! Also, if anyone has a better idea for the tags, please suggest an edit.

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    $\begingroup$ I think the $p$-adic numbers can have analysis on them. $\endgroup$
    – Arthur
    Apr 25, 2018 at 18:29
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    $\begingroup$ Look up $p$-adic analysis. $\endgroup$ Apr 25, 2018 at 18:29
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    $\begingroup$ non-archimedean local fields? $\endgroup$
    – K B Dave
    Apr 25, 2018 at 18:35
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    $\begingroup$ You can do a bit of analysis with the surreal numbers, e.g. power series converge for infinitesimal arguments and certain real-analytic functions can be extended to No. $\endgroup$ Apr 25, 2018 at 18:37
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    $\begingroup$ Fun Fact: the quaternions have very few functions that are differentiable on an open set, just $z \mapsto az + b$. The short reason is that there are too many Cauchy-Riemann equations. $\endgroup$
    – JonathanZ
    Apr 25, 2018 at 18:43

2 Answers 2

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Besides $\mathbb R$ and $\mathbb C$, another family of fields over which one can do mathematical analysis with tools very similar to the usual ones in $\mathbb R$ are the fields $\mathbb Q _p$ of $p$-adic numbers. These numbers and the mathematical analysis done with them are crucially important in number theory, as shown by John Tate in his famous doctoral thesis.

Going even further, one may take a $D$ Dedekind domain and $Q$ its field of fractions. If $P$ is a prime ideal in $D$, then one may consider the completion $Q _P$ of $Q$ under the norm $|q|_P = c ^{-\operatorname{ord} _P (x)}$, where $\operatorname{ord} _P (x)$ is the power of the ideal $P$ in the factorization of the fractional ideal $(x)$ into powers of prime ideals and their inverses, and $c>1$ is arbitrary (any two such numbers $c$ will give different norms but identical topologies).

If one takes $D = \mathbb Z$ in the construction above, and $P = (p)$, one gets $Q_P = \mathbb Q _p$.

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The minimal requirements to have some interesting analysis on a field are:

  • a non trivial metric (i.e. not the discrete topology, which excludes finite fields) for which translations are isometries;

  • completeness, i.e. Cauchy sequences have limits;

  • local compactness (this ensures existence of the Haar measure, a measure invariant by translations which is unique up to a normalization factor).

Mind that connectedness is not too important. For instance the $p$-adic fields are not connected.

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