some structures on ${\mathbb{R}}^{n}$
Let $n\in \{1,2,\mathrm{\dots}\}$. Then, as a set, ${\mathbb{R}}^{n}$ is the $n$fold Cartesian product of the real numbers.
0.0.1 Vector space structure of ${\mathbb{R}}^{n}$
If $u=({u}_{1},\mathrm{\dots},{u}_{n})$ and $v=({v}_{1},\mathrm{\dots},{v}_{n})$ are points in ${\mathbb{R}}^{n}$, we define their sum as
$$u+v=({u}_{1}+{v}_{1},\mathrm{\dots},{u}_{n}+{v}_{n}).$$ 
Also, if $\lambda $ is a scalar (real number), then scalar multiplication is defined as
$$\lambda \cdot u=(\lambda {u}_{1},\mathrm{\dots},\lambda {u}_{n}).$$ 
With these operations^{}, ${\mathbb{R}}^{n}$ becomes a vector space^{} (over $\mathbb{R}$) with dimension^{} $n$. In other words, with this structure^{}, we can talk about, vectors, lines, subspaces^{} of different dimension.
0.0.2 Inner product for ${\mathbb{R}}^{n}$
For $u$ and $v$ as above, we define the inner product as
$$\u27e8u,v\u27e9={u}_{1}{v}_{1}+\mathrm{\cdots}+{u}_{n}{v}_{n}.$$ 
With this product^{}, ${\mathbb{R}}^{n}$ is called an Euclidean space^{}.
We have also an induced norm $\parallel u\parallel =\sqrt{\u27e8u,u\u27e9}$, which gives ${\mathbb{R}}^{n}$ the structure of a normed space (and thus metric space). This inner product let us talk about length, angle between vectors, orthogonal vectors^{}.
0.0.3 Topology for ${\mathbb{R}}^{n}$
The usual topology for ${\mathbb{R}}^{n}$ is the topology^{} induced by the metric
$$d(x,y)=\parallel xy\parallel .$$ 
As a basis for the topology induced by the above norm, one can take open balls $$ where $r>0$ and $x\in {\mathbb{R}}^{n}$.
Properties of the topological space ${\mathbb{R}}^{n}$ are:

1.
${\mathbb{R}}^{n}$ is second countable, i.e., ${\mathbb{R}}^{n}$ has a countable basis.

2.
(HeineBorel theorem) A set in ${\mathbb{R}}^{n}$ is compact^{} if and only if it is closed and bounded^{}.

3.
Since ${\mathbb{R}}^{n}$ is a metric space, ${\mathbb{R}}^{n}$ is a Hausdorff space.
Title  some structures on ${\mathbb{R}}^{n}$ 

Canonical name  SomeStructuresOnmathbbRn 
Date of creation  20130322 14:03:47 
Last modified on  20130322 14:03:47 
Owner  drini (3) 
Last modified by  drini (3) 
Numerical id  10 
Author  drini (3) 
Entry type  Definition 
Classification  msc 54E35 
Classification  msc 53A99 