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


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

  • producto interno
  • ortogonalidad
  • expansión de base
en el contexto de n . Vamos a tomar lo que conocemos sobre vectores y aplicarlo a funciones (señales de tiempo continuo).


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

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

x x 0 x 1 , norma 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

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Para ser una norma, · debe satisfacer:

  • la norma de todo vector es positiva x x S x 0
  • escalando el vector, se escala la norma por la misma cantidad α x α x para todos los vectores x y escalares α
  • Propiedad del Triángulo: x y 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 .


n n ), x x 0 x 1 x n - 1 , 1 x i 0 n 1 x i , n con esta norma es llamado 1 ( [ 0 , n - 1 ] ) .

Colección de todas las x 2 con 1 x 1
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n n ), con norma 2 x i 0 n 1 x i 2 1 2 , n es llamado 2 ( [ 0 , n - 1 ] ) (la usual "norma Euclideana").

Colección de todas las x 2 with 2 x 1
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n (or n , with norm x i x i is called ( [ 0 , n - 1 ] )

x 2 con x 1
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Espacios de secuencias y funciones

Podemos definir normas similares para espacios de secuencias y funciones.

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

  • 1 x n i x i , x n 1 ( ) 1 x
  • 2 x n i x i 2 1 2 , x n 2 ( ) 2 x
  • p x n i x i p 1 p , x n p ( ) p x
  • x n sup i | x [ i ] | , x n ( ) x

Para funciones continuas en el tiempo:

  • p f t t f t p 1 p , f t L p ( ) p f t
  • (En el intervalo) p f t t 0 T f t p 1 p , f t L p ( [ 0 , T ] ) p f t

Questions & Answers

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Source:  OpenStax, Señales y sistemas. OpenStax CNX. Sep 28, 2006 Download for free at http://cnx.org/content/col10373/1.2
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