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# 15.3: Normas

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## Introducción

Este módulo explicará normas, un concepto matemático que proporciona una noción del tamaño de un vector. Específicamente, se discutirá la definición general de una norma y se presentarán normas discretas de señal de tiempo.

## Normas

La norma de un vector es un número real que representa el “tamaño” del vector.

Ejemplo$$\PageIndex{1}$$

En$$\mathbb{R}^2$$, podemos definir una norma para ser una longitud geométrica de vectores.

$$\boldsymbol{x}=(x_0,x_1)^T$$, norma$$\|\boldsymbol{x}\|=\sqrt{x_{0}^{2}+x_{1}^{2}}$$

Matemáticamente, una norma$$\|\cdot\|$$ es solo una función (tomar un vector y devolver un número real) que satisface tres reglas.

Para ser una norma,$$\|\cdot\|$$ debe satisfacer:

1. la norma de cada vector es positiva$$\|x\|>0$$,$$x \in S$$
2. escalar un vector escala la norma por la misma cantidad$$\|\alpha x\|=|\alpha|\|x\|$$ para todos los vectores$$x$$ y escalares$$\alpha$$
3. Triángulo Propiedad:$$\|x+y\| \leq\|x\|+\|y\|$$ 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 (Sección 15.2) con una norma bien definida se denomina espacio vectorial normado o espacio lineal normado.

### Ejemplos

Ejemplo$$\PageIndex{2}$$

$$\mathbb{R}^n$$(o$$\mathbb{C}^n$$),\ (\ negridsymbol {x} =\ left (\ begin {array} {c}
x_ {0}\\
x_ {1}\
\\ puntos\\
x_ {n-1}
\ end {array}\ derecha)\)$$\|x\|_{1}=\sum_{i=0}^{n-1}\left|x_{i}\right|$$,$$\mathbb{R}^n$$ con esta norma se llama$$\ell^{1}([0, n-1])$$.

Ejemplo$$\PageIndex{3}$$

$$\mathbb{R}^n$$(o$$\mathbb{C}^n$$), con norma$$\|x\|_{2}=\left(\sum_{i=0}^{n-1}\left(\left|x_{i}\right|\right)^{2}\right)^{\frac{1}{2}}$$,$$\mathbb{R}^n$$ se llama$$\ell^{2}([0, n-1])$$ (la habitual norma “euclidiana”).

Ejemplo$$\PageIndex{4}$$

$$\mathbb{R}^n$$(o$$\mathbb{C}^n$$), con norma$$\|x\|_{\infty}=\max _{i}\left\{i,\left|x_{i}\right|\right\}$$ se llama$$\ell^{\infty}([0, n-1])$$

### Espacios de Secuencias y Funciones

Podemos definir normas similares para espacios de secuencias y funciones.

Señales de tiempo discretas = secuencias de números

$x[n]=\left\{\ldots, x_{-2}, x_{-1}, x_{0}, x_{1}, x_{2}, \ldots\right\} \nonumber$

• $$\|x(n)\|_{1}=\sum_{i=-\infty}^{\infty}|x[i]|, x[n] \in \ell^{1}(\mathbb{Z}) \Rightarrow\left(\|x\|_{1}<\infty\right)$$
• $$\|x(n)\|_{2}=\left(\sum_{i=-\infty}^{\infty}(|x[i]|)^{2}\right)^{\frac{1}{2}}, x[n] \in \ell^{2}(\mathbb{Z}) \Rightarrow\left(\|x\|_{2}<\infty\right)$$
• $$\|x(n)\|_{p}=\left(\sum_{i=-\infty}^{\infty}(|x[i]|)^{P}\right)^{\frac{1}{p}}, x[n] \in \ell^{p}(\mathbb{Z}) \Rightarrow\left(\|x\|_{p}<\infty\right)$$
• $$\|x(n)\|_{\infty}=\sup _{i}|x[i]|, x[n] \in \ell^{\infty}(\mathbb{Z}) \Rightarrow\left(\|x\|_{\infty}<\infty\right)$$

Para funciones de tiempo continuo:

• $$\|f(t)\|_{p}=\left(\int_{-\infty}^{\infty}(|f(t)|)^{p} d t\right)^{\frac{1}{p}}, f(t) \in L^{p}(\mathbb{R}) \Rightarrow\left(\|f(t)\|_{p}<\infty\right)$$
• $$\|f(t)\|_{p}=\left(\int_{0}^{T}(|f(t)|)^{p} d t\right)^{\frac{1}{p}}, f(t) \in L^{p}([0, T]) \Rightarrow\left(\|f(t)\|_{p}<\infty\right)$$

This page titled 15.3: Normas is shared under a CC BY license and was authored, remixed, and/or curated by Richard Baraniuk et al..