Introduction to Mathematical Physics/Vectorial spaces

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Definition[edit]

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 space[edit]

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.