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

*

*$d(x,x)=0$


*$d(x,y)>0$


*$d(x,y)=d(y,x)$


*$d(x,z)\leq d(x,y)+d(y,z)$ (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!
