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Normas

Module by: Michael Haag, Justin Romberg. E-mail the authorsTranslated By: Fara Meza, Erika Jackson

Based on: Norms by Michael Haag, Justin Romberg

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

Introducción

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

en el contexto de Rn 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.

Ejemplo 1

En R2 2 , podemos definir la norma que sea la longitud geométrica de los vectores.

Figura 1
Figura 1 (norm_f1.png)

x= x 0 x 1 T x x 0 x 1 , norma x= x 0 2+ x 1 2 x 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:

  1. la norma de todo vector es positiva x>0  ,   xS    x x S x 0
  2. escalando el vector, se escala la norma por la misma cantidad αx=|α|x α x α x para todos los vectores x x y escalares α α
  3. Propiedad del Triángulo: x+yx+y x y x y para todos los vectores x x, y 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

Ejemplo 2

Rn n Cn n ), x= x 0 x 1 x n - 1 x x 0 x 1 x n - 1 , x1=i=0n1| x i | 1 x i 0 n 1 x i , Rn n con esta norma es llamado 1 ( [ 0 , n - 1 ] ) 1 ( [ 0 , n - 1 ] ) .

Figura 2: Colección de todas las xR2 x 2 con x1=1 1 x 1
Figura 2 (norm_f2.png)

Ejemplo 3

Rn n Cn n ), con norma x2=i=0n1| x i |212 2 x i 0 n 1 x i 2 1 2 , Rn n es llamado 2 ( [ 0 , n - 1 ] ) 2 ( [ 0 , n - 1 ] ) (la usual "norma Euclideana").

Figura 3: Colección de todas las xR2 x 2 with x2=1 2 x 1
Figura 3 (norm_f3.png)

Ejemplo 4

Rn n (or Cn n , with norm x=maxii| x i | x i x i is called ( [ 0 , n - 1 ] ) ( [ 0 , n - 1 ] )

Figura 4: xR2 x 2 con x=1 x 1
Figura 4 (norm_f4.png)

Espacios de Secuencias y Funciones

Podemos definir normas similares para espacios de secuencias y funciones.

Señales de tiempo discreto= secuencia de números xn= x -2 x -1 x 0 x 1 x 2 x n x -2 x -1 x 0 x 1 x 2

  • xn1=i=|xi| 1 x n i x i , xn 1 ( ) (x1<) x n 1 ( ) 1 x
  • xn2=i=|xi|212 2 x n i x i 2 1 2 , xn 2 ( ) (x2<) x n 2 ( ) 2 x
  • xnp=i=|xi|p1p p x n i x i p 1 p , xn p ( ) (xp<) x n p ( ) p x
  • xn= sup i | x [ i ] | x n sup i | x [ i ] | , xn ( ) (x<) x n ( ) x

Para funciones continuas en el tiempo:

  • ftp=|ft|pdt1p p f t t f t p 1 p , ft L p ( ) (ftp<) f t L p ( ) p f t
  • (En el intervalo) ftp=0T|ft|pdt1p p f t t 0 T f t p 1 p , ft L p ( [ 0 , T ] ) (ftp<) f t L p ( [ 0 , T ] ) p f t

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