# 7.1 Normas

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Este modulo definirá una norma y da unos ejemplos y sus propiedades.

## IntroducciÓN

Mucho del lenguaje utilizado en esta sección seráfamiliar para usted- debe de haber estado expuesto a los conceptos de

• producto interno
• expansión de base
en el contexto de $\mathbb{R}^{n}$ . Vamos a tomar lo que conocemos sobre vectores y aplicarlo a funciones (señales de tiempo continuo).

## Normas

La norma de un vector es un número real que representa el "tamaño" de el vector.

En $\mathbb{R}^{2}$ , podemos definir la norma que sea la longitud geométrica de los vectores.

$x=\left(\begin{array}{c}{x}_{0}\\ {x}_{1}\end{array}\right)$ , norma $(x)=\sqrt{{x}_{0}^{2}+{x}_{1}^{2}}$

Matemáticamente, una norma $(·)$ es solo una función (tomando un vector y regresando un número real) que satisface tres reglas

Para ser una norma, $(·)$ debe satisfacer:

• la norma de todo vector es positiva $\forall x, x\in S\colon (x)> 0$
• escalando el vector, se escala la norma por la misma cantidad $(\alpha x)=\left|\alpha \right|(x)$ para todos los vectores $x$ y escalares $\alpha$
• Propiedad del Triángulo: $(x+y)\le (x)+(y)$ para todos los vectores $x$ , $y$ .“El“tamaño“de la suma de dos vectores es menor o igual a la suma de sus tamaños”

Un espacio vectorial con una norma bien definida es llamado un espacio vectorial normado o espacio lineal normado .

## Ejemplos

$\mathbb{R}^{n}$ $\mathbb{C}^{n}$ ), $x=\left(\begin{array}{c}{x}_{0}\\ {x}_{1}\\ \dots \\ {x}_{n-1}\end{array}\right)$ , $(, x)=\sum_{i=0}^{n-1} \left|{x}_{i}\right|$ , $\mathbb{R}^{n}$ con esta norma es llamado ${\ell }^{1}\left(\left[0,n-1\right]\right)$ .

$\mathbb{R}^{n}$ $\mathbb{C}^{n}$ ), con norma $(, x)=\sum_{i=0}^{n-1} \left|{x}_{i}\right|^{2}^{\left(\frac{1}{2}\right)}$ , $\mathbb{R}^{n}$ es llamado ${\ell }^{2}\left(\left[0,n-1\right]\right)$ (la usual "norma Euclideana").

$\mathbb{R}^{n}$ (or $\mathbb{C}^{n}$ , with norm $()$ x i x i is called ${\ell }^{\infty }\left(\left[0,n-1\right]\right)$

## Espacios de secuencias y funciones

Podemos definir normas similares para espacios de secuencias y funciones.

Señales de tiempo discreto= secuencia de números $x(n)=\{\dots , {x}_{-2}, {x}_{-1}, {x}_{0}, {x}_{1}, {x}_{2}, \dots \}$

• $(, x(n))=\sum_{i=()}$ x i , $x(n)\in {\ell }^{1}\left(ℤ\right)\implies ((, x))$
• $(, x(n))=\sum_{i=()} ^{}$ x i 2 1 2 , $x(n)\in {\ell }^{2}\left(ℤ\right)\implies ((, x))$
• $(, x(n))=\sum_{i=()} ^{}$ x i p 1 p , $x(n)\in {\ell }^{p}\left(ℤ\right)\implies ((, x))$
• $()$ x n sup i | x [ i ] | , $x(n)\in {\ell }^{\infty }\left(ℤ\right)\implies (())$ x

Para funciones continuas en el tiempo:

• $(, f(t))=\int_{()} \,d t^{}$ f t p 1 p , $f(t)\in {L}^{p}\left(ℝ\right)\implies ((, f(t)))$
• (En el intervalo) $(, f(t))=\int_{0}^{T} \left|f(t)\right|^{p}\,d t^{\left(\frac{1}{p}\right)}$ , $f(t)\in {L}^{p}\left(\left[0,T\right]\right)\implies ((, f(t)))$

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