Introduction to Mathematical Physics/Vectorial spaces

DefinitionEdit

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

Definition:

An ensemble   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:


  is a commutative group.

    where   is the unity of   law.

 



Functional spaceEdit

Definition:

A functional space is a set   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   of   is a mapping from   into  .

  designs the number associated to function   by functional  .

Definition:

A functional   is linear if for any functions   and   of   and any complex numbers   and   :

 

Definition:

Space   is the vectorial space of functions indefinitely derivable with a bounded support.