# 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.