Introduction to Mathematical Physics/Vectorial spaces

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Definition[edit]

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[edit]

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.