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 $C$.

$<T|\phi >$ 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 $\lambda _{2}$ :

$<T|\lambda _{1}\phi _{1}+\lambda _{2}\phi _{2}>=\lambda _{1}<T|\phi _{1}>+\lambda _{2}<T|\phi _{2}>$

Definition:

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