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Introduction to Mathematical Physics/Vectorial spaces
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Definition
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.
Functional space
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.