Let E be a complex vector space. A norm on E is given by the following definition
Norm
A norm ∥.∥ is a real positive function such that
∀x∈E,∥x∥≥0∀x∈E,∥x∥=0⇔x=0∀λ∈C,∥λx∥=∣λ∣∥x∥∀x,y∈E,∥x+y∥≤∥x∥+∥y∥
The most used norms are
1-Norm
Let f:R→C, ∥f∥1=∫R∣f(t)∣dt
Let u a real or complex sequence, ∥u∥1=n=−∞∑+∞∣un∣
2-Norm (squared root of Energy)
Soit f:R→C, ∥f∥2=∫R∣f(t)∣2dt
Soit u une suite réelle ou complexe, ∥u∥2=n=−∞∑+∞∣un∣2
∞-Norm
Soit f:R→C, ∥f∥∞=tsup∣f(t)∣
Soit u une suite réelle ou complexe, ∥u∥∞=nsup∣un∣
p-Norm
Soit f:R→C, ∥f∥p=(∫R∣f(t)∣pdt)1/p
Soit u une suite réelle ou complexe, ∥u∥p=(n=−∞∑+∞∣un∣p)1/p
On peut alors définir des espaces vectoriels normés couramment
utilisés:
Espace Lp(E)
The set of functions f of E with a finite p-Norm is denoted by Lp(E)
In particular, we will be interested by the space L2(R) of functions with a finite energy. We can define similarly the space ℓp(Z) of sequences with a finite energy.
A norms allows one to define the notion of distance. Moreover, a norm allows one to define the notion of convergence.
Convergence in norm
Let E be a complex vector space with the norm ∥.∥. A sequence
fnn∈N of E converges to f∈E
iff
n→+∞lim∥fn−f∥=0.
we will denote by
n→+∞limfn=f
The notion of convergence depends then of the chosen norm. The three most usefull convergences are
The uniform convergence, with the ∞-Norm.
The absolute convergence, with the 1-Norm
The convergence in energy, with the 2-Norm
Finally, a vector space E with a norm is said to be a Banach space, is every Cauchy’s sequence is a convergent sequence.