Norms and convergence

Let EE be a complex vector space. A norm on EE is given by the following definition

Norm A norm .\Vert.\Vert is a real positive function such that xE, x0\forall x\in E, \ \Vert x\Vert \geq 0 xE, x=0x=0\forall x\in E, \ \Vert x\Vert = 0 \Leftrightarrow x = 0 λC, λx=λx\forall \lambda\in\mathbb{C},\ \Vert \lambda x\Vert = |\lambda| \Vert x\Vert x,yE, x+yx+y\forall x,y\in E,\ \Vert x+y\Vert\leq \Vert x\Vert + \Vert y\Vert

The most used norms are

  • 1-Norm
    • Let f:RCf:\mathbb{R}\rightarrow\mathbb{C}, f1=Rf(t)dt\Vert f\Vert_1 = \int_{\mathbb{R}}{ |f(t)| \mathrm{d} t}
    • Let uu a real or complex sequence, u1=n=+un\Vert u\Vert_1 = \sum\limits_{n=-\infty}^{+\infty} |u_n|
  • 2-Norm (squared root of Energy)
    • Soit f:RCf:\mathbb{R}\rightarrow\mathbb{C}, f2=Rf(t)2dt\Vert f\Vert_2 = \sqrt{\int_\mathbb{R} |f(t)|^2 \mathrm{d} t}
    • Soit uu une suite réelle ou complexe, u2=n=+un2\Vert u\Vert_2 =\sqrt{\sum\limits_{n=-\infty}^{+\infty} |u_n|^2}
  • \infty-Norm
    • Soit f:RCf:\mathbb{R}\rightarrow\mathbb{C}, f=suptf(t)\Vert f\Vert_\infty = \sup\limits_t |f(t)|
    • Soit uu une suite réelle ou complexe, u=supnun\Vert u\Vert_\infty = \sup\limits_{n} |u_n|
  • pp-Norm
    • Soit f:RCf:\mathbb{R}\rightarrow\mathbb{C}, fp=(Rf(t)pdt)1/p\Vert f\Vert_p = \left(\int_\mathbb{R} |f(t)|^p \mathrm{d} t\right)^{1/p}
    • Soit uu une suite réelle ou complexe, up=(n=+unp)1/p\Vert u\Vert _p =\left(\sum\limits_{n=-\infty}^{+\infty} |u_n|^p\right)^{1/p}

On peut alors définir des espaces vectoriels normés couramment utilisés:

Espace Lp(E)L^p(E) The set of functions ff of EE with a finite pp-Norm is denoted by Lp(E)L^p(E)

In particular, we will be interested by the space L2(R)L^2(\mathbb{R}) of functions with a finite energy. We can define similarly the space p(Z)\ell^p(\mathbb{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 EE be a complex vector space with the norm .\Vert .\Vert . A sequence fnnN{f_n}_{n\in\mathbb{N}} of EE converges to fEf\in E iff limn+fnf=0 . \lim_{n\rightarrow +\infty}\Vert f_n -f\Vert = 0\ . we will denote by limn+fn=f \lim_{n\rightarrow +\infty} f_n = f

The notion of convergence depends then of the chosen norm. The three most usefull convergences are

  • The uniform convergence, with the \infty-Norm.
  • The absolute convergence, with the 1-Norm
  • The convergence in energy, with the 2-Norm

Finally, a vector space EE with a norm is said to be a Banach space, is every Cauchy’s sequence is a convergent sequence.