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 | ϕ > designs the number associated to function ϕ by functional T.

Definition:

A functional T is linear if for any functions ϕ₁ and ϕ₂ of ${\mathcal F}$ and any complex numbers λ₁ and λ₂ :

< T | λ₁ϕ₁ + λ₂ϕ₂ > = λ₁ < T | ϕ₁ > + λ₂ < T | ϕ₂ >

Definition:

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