some structures on $\mathbbmss{R}^{n}$

Let $n\in\{1,2,\ldots\}$. Then, as a set, $\mathbbmss{R}^{n}$ is the $n$-fold Cartesian product of the real numbers.

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

If $u=(u_{1},\ldots,u_{n})$ and $v=(v_{1},\ldots,v_{n})$ are points in $\mathbbmss{R}^{n}$, we define their sum as

 $u+v=(u_{1}+v_{1},\ldots,u_{n}+v_{n}).$

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

 $\lambda\cdot u=(\lambda u_{1},\ldots,\lambda u_{n}).$

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

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

 $\langle u,v\rangle=u_{1}v_{1}+\cdots+u_{n}v_{n}.$

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

We have also an induced norm $\left\|u\right\|=\sqrt{\langle u,u\rangle}$, which gives $\mathbbmss{R}^{n}$ 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 $\mathbbmss{R}^{n}$

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

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

As a basis for the topology induced by the above norm, one can take open balls $B(x,r)=\{y\in\mathbbmss{R}^{n}\mid\left\|x-y\right\| where $r>0$ and $x\in\mathbbmss{R}^{n}$.

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

1. 1.

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

2. 2.

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

3. 3.

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

Title some structures on $\mathbbmss{R}^{n}$ SomeStructuresOnmathbbRn 2013-03-22 14:03:47 2013-03-22 14:03:47 drini (3) drini (3) 10 drini (3) Definition msc 54E35 msc 53A99