# Introduction to Mathematical Physics/Vectorial spaces

## Definition

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

Definition:

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

$(E,+)$  is a commutative group.

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

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

## Functional space

Definition:

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

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

Definition:

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

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

Definition:

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