Norms and convergence

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

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

The most used norms are

  • 1-Norm
    • Let $f:\mathbb{R}\rightarrow\mathbb{C}$, $\Vert f\Vert_1 = \int_{\mathbb{R}}{ |f(t)| \mathrm{d} t}$
    • Let $u$ a real or complex sequence, $\Vert u\Vert_1 = \sum\limits_{n=-\infty}^{+\infty} |u_n|$
  • 2-Norm (squared root of Energy)
    • Soit $f:\mathbb{R}\rightarrow\mathbb{C}$, $\Vert f\Vert_2 = \sqrt{\int_\mathbb{R} |f(t)|^2 \mathrm{d} t}$
    • Soit $u$ une suite réelle ou complexe, $\Vert u\Vert_2 =\sqrt{\sum\limits_{n=-\infty}^{+\infty} |u_n|^2}$
  • $\infty$-Norm
    • Soit $f:\mathbb{R}\rightarrow\mathbb{C}$, $\Vert f\Vert_\infty = \sup\limits_t |f(t)|$
    • Soit $u$ une suite réelle ou complexe, $\Vert u\Vert_\infty = \sup\limits_{n} |u_n|$
  • $p$-Norm
    • Soit $f:\mathbb{R}\rightarrow\mathbb{C}$, $\Vert f\Vert_p = \left(\int_\mathbb{R} |f(t)|^p \mathrm{d} t\right)^{1/p}$
    • Soit $u$ une suite réelle ou complexe, $\Vert u\Vert _p =\left(\sum\limits_{n=-\infty}^{+\infty} |u_n|^p\right)^{1/p}$

On peut alors définir des espaces vectoriels normés couramment utilisés:

Espace $L^p(E)$ The set of functions $f$ of $E$ with a finite $p$-Norm is denoted by $L^p(E)$

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

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 $E$ with a norm is said to be a Banach space, is every Cauchy’s sequence is a convergent sequence.