# Introduction to Mathematical Physics/Vectorial spaces

## Definition

Let ${\displaystyle K}$  be ${\displaystyle R}$  of ${\displaystyle C}$ . An ensemble ${\displaystyle E}$  is a vectorial space if it has an algebric structure defined by to laws ${\displaystyle +}$  and ${\displaystyle .}$ , such that every linear combination of two elements of ${\displaystyle E}$  is inside ${\displaystyle E}$ . More precisely:

Definition:

An ensemble ${\displaystyle E}$  is a vectorial space if it has an algebric structure defined by to laws, a composition law noted ${\displaystyle +}$  and an action law noted ${\displaystyle .}$ , those laws verifying:

${\displaystyle (E,+)}$  is a commutative group.

${\displaystyle \forall (\alpha ,\beta )\in K\times K,\forall x\in E\alpha (\beta x)=(\alpha \beta )x}$  ${\displaystyle \forall x\in E,1.x=x}$  where ${\displaystyle 1}$  is the unity of ${\displaystyle .}$  law.

${\displaystyle \forall (\alpha ,\beta )\in K\times K,\forall (x,y)\in E\times E(\alpha +\beta )x=\alpha x+\beta x{\mbox{ and }}\alpha (x+y)=\alpha x+\alpha y}$

## Functional space

Definition:

A functional space is a set ${\displaystyle {\mathcal {F}}}$  of functions that have a vectorial space structure.

The set of the function continuous on an interval is a functional space. The set of the positive functions is not a fucntional space.

Definition:

A functional ${\displaystyle T}$  of ${\displaystyle {\mathcal {F}}}$  is a mapping from ${\displaystyle {\mathcal {F}}}$  into ${\displaystyle C}$ .

${\displaystyle }$  designs the number associated to function ${\displaystyle \phi }$  by functional ${\displaystyle T}$ .

Definition:

A functional ${\displaystyle T}$  is linear if for any functions ${\displaystyle \phi _{1}}$  and ${\displaystyle \phi _{2}}$  of ${\displaystyle {\mathcal {F}}}$  and any complex numbers ${\displaystyle \lambda _{1}}$  and ${\displaystyle \lambda _{2}}$  :

${\displaystyle =\lambda _{1}+\lambda _{2}}$

Definition:

Space ${\displaystyle {\mathcal {D}}}$  is the vectorial space of functions indefinitely derivable with a bounded support.