some structures on ${ℝ}^n$

Let $n\in \{1,2,\mathrm{\dots }\}$. Then, as a set, ${ℝ}^n$ is the $n$-fold Cartesianproduct of the real numbers.

0.0.1 Vector space structure of ${ℝ}^n$

If $u=(u_1,\mathrm{\dots },u_n)$ and$v=(v_1,\mathrm{\dots },v_n)$ are points in ${ℝ}^n$, we define their sumas

$$u+v=(u_1+v_1,\mathrm{\dots },u_n+v_n).$$

Also, if $\lambda$ is a scalar (real number), then scalar multiplicationis defined as

$$\lambda \cdot u=(\lambda u_1,\mathrm{\dots },\lambda u_n).$$

With these operations, ${ℝ}^n$ becomes a vector space (over $ℝ$) 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 ${ℝ}^n$

For $u$ and $v$ as above, we define the inner product as

$$⟨u,v⟩=u_1{v}_1+\mathrm{\cdots }+u_n{v}_n.$$

With this product, ${ℝ}^n$ is called an Euclidean space.

We have also an induced norm $\parallel u\parallel =\sqrt{⟨u,u⟩}$, which gives${ℝ}^n$ 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 ${ℝ}^n$

The usual topology for ${ℝ}^n$ is the topology induced by the metric

$$d(x,y)=\parallel x-y\parallel .$$

As a basis for the topology induced by the above norm, one can takeopen balls where $r>0$and $x\in {ℝ}^n$.

Properties of the topological space ${ℝ}^n$ are:

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

2. (Heine-Borel theorem)A set in ${ℝ}^n$ is compact if and only if it is closed and bounded.

3. Since ${ℝ}^n$ is a metric space, ${ℝ}^n$ is a Hausdorff space.