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Normas

Module by: Michael Haag, Justin Romberg Translated By Fara Meza, Erika JacksonBased on: Norms by Michael Haag, Justin Romberg

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

Note: Your browser may not currently support MathML. See our browser support page for additional details. You can always view the correct math in the PDF version.

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,x∈S:∥x∥>0 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+y∥≤∥x∥+∥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 (ó ℂn n ), x= x 0 x 1 … x n - 1 x x 0 x 1 … x n - 1 , ∥x∥1=∑i=0n−1| x i | 1 x i 0 n 1 x i , ℝn n con esta norma es llamado ℓ 1 ( [ 0 , n - 1 ] ) ℓ 1 ( [ 0 , n - 1 ] ) .

Figura 2: Colección de todas las x∈ℝ2 x 2 con ∥x∥1=1 1 x 1
Figura 2 (norm_f2.png)

Ejemplo 3

ℝn n (ó ℂn n ), con norma ∥x∥2=∑i=0n−1| x i |212 2 x i 0 n 1 x i 2 1 2 , ℝn n es llamado ℓ 2 ( [ 0 , n - 1 ] ) ℓ 2 ( [ 0 , n - 1 ] ) (la usual "norma Euclideana").

Figura 3: Colección de todas las x∈ℝ2 x 2 with ∥x∥2=1 2 x 1
Figura 3 (norm_f3.png)

Ejemplo 4

ℝn n (or ℂn n , with norm ∥x∥∞=maxi{| x i |} x i x i is called ℓ ∞ ( [ 0 , n - 1 ] ) ℓ ∞ ( [ 0 , n - 1 ] )

Figura 4: x∈ℝ2 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 …

  • ∥xn∥1=∑i=-∞∞|xi| 1 x n i x i , xn∈ ℓ 1 ( ℤ ) ⇒∥x∥1<∞ x n ℓ 1 ( ℤ ) 1 x
  • ∥xn∥2=∑i=-∞∞|xi|212 2 x n i x i 2 1 2 , xn∈ ℓ 2 ( ℤ ) ⇒∥x∥2<∞ x n ℓ 2 ( ℤ ) 2 x
  • ∥xn∥p=∑i=-∞∞|xi|p1p p x n i x i p 1 p , xn∈ ℓ p ( ℤ ) ⇒∥x∥p<∞ x n ℓ p ( ℤ ) p x
  • ∥xn∥∞= sup i | x [ i ] | x n sup i | x [ i ] | , xn∈ ℓ ∞ ( ℤ ) ⇒∥x∥∞<∞ x n ℓ ∞ ( ℤ ) x

Para funciones continuas en el tiempo:

  • ∥ft∥p=∫-∞∞|ft|pdt1p p f t t f t p 1 p , ft∈ L p ( ℝ ) ⇒∥ft∥p<∞ f t L p ( ℝ ) p f t
  • (En el intervalo) ∥ft∥p=∫0T|ft|pdt1p p f t t 0 T f t p 1 p , ft∈ L p ( [ 0 , T ] ) ⇒∥ft∥p<∞ f t L p ( [ 0 , T ] ) p f t

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