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:
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.
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.
A functional of is a mapping from into .
designs the number associated to function by functional .
A functional is linear if for any functions and of and any complex numbers and :
Space is the vectorial space of functions indefinitely derivable with a bounded support.