Introduction to Mathematical Physics/Vectorial spaces

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

Functional space
The set of the function continuous on an interval is a functional space. The set of the positive functions is not a fucntional space.

$$$$ designs the number associated to function $$\phi$$ by functional $$T$$.