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?