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A space is an ordered tuple, where the first element is a set and the following elements are describing the added structure, e.g. $(X, m)$ for a metric space, $(X, \tau)$ for a topological space. What are the following elements for a Euclidean space?

As far as I understand we need

  • $X=\mathbb R^n$ is the set of all n-tuples of real numbers (with $n\in\mathbb N$)
  • we need the elements of $X$ to be vectors - so linearly combine-able with the scalar multiplication $\times$, the field $F$ and addition $+$.
  • a dot product $\cdot$ between the elements of $X$.
  • a norm for the elements of $X$. Is this inherently included in the dot product or do I need to state it explicitly to be precise? Don't I need an additional "$-$"? http://faculty.cord.edu/ahendric/2008Fall210/subsub.pdf suggests that this is also included in the "$+$".
  • completeness of $X$ (is this inherently included in the fact that $X=\mathbb R^n$?)
  • a metric (I think this is also inherently included in the norm and the fact that the elements of $X$ are vectors, right?)

From that I infer, that a Euclidean space is $(\mathbb R^n, \cdot, +, F, \times)$. Possibly I also need a "$-$".

So: How do I formally write down a Euclidean space with symbols?

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  • $\begingroup$ Simply $\mathbb R^n$ may suffice, because the addition, product, norm, etc. are standard $\endgroup$
    – J. W. Tanner
    Nov 27, 2020 at 15:38
  • $\begingroup$ I am not sure what your $F$ is supposed to be. As the other comment says, the operations on $\Bbb{R}^n$ are standard but if you want to list a primitive set of operation, then: (1) you do need the ring (or field) operations on $\Bbb{R}$, but from your metric space example, it looks like you are taking those as given and (2) there is no canonical choice e.g., a norm (satisfying an appropriate axiom) and a dot product are interdefinable. Negation can be axiomatised using addition. Alternatively addition can be defined using subtraction and negation. $\endgroup$
    – Rob Arthan
    Nov 27, 2020 at 15:39
  • $\begingroup$ @RobArthan: The $F$ is the field (see the second bullet point). $\endgroup$
    – Make42
    Nov 27, 2020 at 15:41
  • $\begingroup$ I missed that, possibly because your first bullet implied the field was $\Bbb{R}$. Thanks. $\endgroup$
    – Rob Arthan
    Nov 27, 2020 at 15:42
  • $\begingroup$ @J.W.Tanner: I want to build a new space $(X, \Xi, ...)$ where $\Xi$ is a special finite set of disjoint subsets of $X$ (not a topology or anything like that). $\endgroup$
    – Make42
    Nov 27, 2020 at 15:43

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You already wrote down a Euclidean space in your question: $\mathbb{R}$.

The only other thing I can think of that you might want to include is your metric. Say $(\mathbb{R},d)$ is a metric space and define d, which is the distance of any two points.

There are some axioms to remember for metrics:

  1. $d(x,x)=0$

  2. $d(x,y)>0$

  3. $d(x,y)=d(y,x)$

  4. $d(x,z)\leq d(x,y)+d(y,z)$ (called the triangle inequality; think of a right triangle, and you walk in a diagonal line to get where you need to go)

There are many metrics we could define for a space like $\mathbb{R^2}$, the real plane; the most common being $\sqrt{(x_1-x_2)^2+(y_1-y_2)^2}$

EDIT:

You would need to learn some topology I suppose. The Cartesian product is just one example of a more general concept which is product spaces. In topology we discuss continuity and open sets (they are not all defined the same). Say $X,Y$ are topological spaces, and the set, $U_{X_i}$ and $V_{Y_i}$ are open in their respective topologies.

We define the topology on the product space $X\,\,x\,\, V$ by just saying it "inherits" the topology of the other two spaces. A subset of $X\,\,x\,\, V$ is open if an only if $U\subset X$ and $V\subset Y$ are both open. This applies exactly the same way to our standard metric spaces, but instead the product space will inherit the metric, which can be thought of as giving us an idea of what "open" is as well!

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    $\begingroup$ From my understanding, saying the $\mathbb R^n$ is the Euclidean space is an abuse of notion, but since I want to build a new space math.stackexchange.com/questions/3925036/…, I need the explicit symbols. Without the abuse of notation I understand that $\mathbb R^n = \mathbb R \times ... \times \mathbb R$ is just the set of n-tuples of real numbers (or the respective Cartesian product $\times$ of reals). Also, did you omit ${}^n$ on purpose? $\endgroup$
    – Make42
    Nov 27, 2020 at 15:51
  • $\begingroup$ @Make42, alright, then you would need to learn a little topology to do product spaces, but it's very easy. $\endgroup$
    – Algebraic
    Nov 27, 2020 at 16:18
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    $\begingroup$ I guess you would need to point me to some specific text in order to get me going. Otherwise I am not sure I understand what you are referring to. (Disclaimer: I am not a pure mathematician, I am "just" an engineer doing applied math in daily life.) $\endgroup$
    – Make42
    Nov 27, 2020 at 16:21
  • $\begingroup$ @Make42 Check my edit. $\endgroup$
    – Algebraic
    Nov 27, 2020 at 16:25
  • $\begingroup$ That was helpful - I understand that part of a product space is that they inherit the topology of the other two spaces. Also - which I was not aware of - it seems (en.wikipedia.org/wiki/Real_number#Definition) that the reals are not just the set of numbers itself (what I thought), but also the "stuff that you can do with them", namely $+,\cdot,<$. But I do not yet see how this directly leads to $\cdot,+,F,\times$ for the elements of the product space. E.g. I use a totally different dot product (which is done with the kernel trick). $\endgroup$
    – Make42
    Nov 27, 2020 at 16:43

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