# Introduction to Mathematical Physics/Vectorial spaces

## 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.