# 10.1 Convergencia de vectores

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Este módulo presenta dos tipos comunes de convergencia, puntual y norma, discutiremos sus propiedades, diferencias y relaciones entre ellos.

## Convergencia de vectores

Discutiremos la convergencia puntual y de la norma de vectores. También existen otros tipos de convergencia y uno en particular, la convergencia uniforme , también puede ser estudiada. Para esta discusión, asumiremos que los vectores pertenecen a un espacio de vector normado .

## Convergencia puntual

Una secuencia $(n, \left\{1\right\})$ g n converge puntualmente al límite $g$ si cada elemento de ${g}_{n}$ converge al elemento correspondiente en $g$ . A continuación hay unos ejemplos para tratar de ilustrar esta idea.

${g}_{n}=\left(\begin{array}{c}{g}_{n}(1)\\ {g}_{n}(2)\end{array}\right)()=\left(\begin{array}{c}1+\frac{1}{n}\\ 2-\frac{1}{n}\end{array}\right)()$ Primero encontramos los siguiente limites para nuestras dos ${g}_{n}$ 's: $\lim_{n\to }n\to$ g n 1 1 $\lim_{n\to }n\to$ g n 2 2 Después tenemos el siguiente, $\lim_{n\to }n\to$ g n g puntual, donde $g=\left(\begin{array}{c}1\\ 2\end{array}\right)$ .

$\forall t, t\in \mathbb{R}\colon {g}_{n}(t)=\frac{t}{n}$ Como se hizo anteriormente, primero examinamos el límite $\lim_{n\to }n\to$ g n t 0 n t 0 n 0 donde ${t}_{0}\in \mathbb{R}$ . Por lo tanto $\lim_{n\to }n\to$ g n g puntualmente donde $g(t)=0$ para toda $t\in \mathbb{R}$ .

## Norma de convergencia

La secuencia $(n, \left\{1\right\})$ g n converge a $g$ en la norma si $\lim_{n\to }n\to$ g n g 0 . Aqui $(˙)$ es la norma del espacio vectorial correspondiente de ${g}_{n}$ 's. Intuitivamente esto significa que la distancia entre los vectores ${g}_{n}$ y $g$ decrese a $0$ .

${g}_{n}=\left(\begin{array}{c}1+\frac{1}{n}\\ 2-\frac{1}{n}\end{array}\right)$ Sea $g=\left(\begin{array}{c}1\\ 2\end{array}\right)$

$({g}_{n}-g)=\sqrt{(1+\frac{1}{n}-1)^{2}+(2-\frac{1}{n})^{2}}=\sqrt{\frac{1}{n^{2}}+\frac{1}{n^{2}}}=\frac{\sqrt{2}}{n}$
Asi $\lim_{n\to }n\to$ g n g 0 , Por lo tanto, ${g}_{n}\to g$ en la norma.

${g}_{n}(t)=\begin{cases}\frac{t}{n} & \text{if 0\le t\le 1}\\ 0 & \text{otherwise}\end{cases}$ Sea $g(t)=0$ para todo $t$ .

$({g}_{n}(t)-g(t))=\int_{0}^{1} \frac{t^{2}}{n^{2}}\,d t=(n, \left[0 , 1\right], \frac{t^{3}}{3n^{2}})=\frac{1}{3n^{2}}$
Asi $\lim_{n\to }n\to$ g n t g t 0 Por lo tanto, ${g}_{n}(t)\to g(t)$ en la norma.

## Puntual vs.norma de convergencia

Para $\mathbb{R}^{m}$ , la convergencia puntual y la norma de convergencia es equivalente.

## Puntual ⇒ norma

${g}_{n}(i)\to g(i)$ Asumiendo lo anterior, entonces $({g}_{n}-g)^{2}=\sum_{i=1}^{m} ({g}_{n}(i)-g(i))^{2}$ Así,

$\lim_{n\to }n\to$ g n g 2 n i m 1 g n i g i 2 i m 1 n g n i g i 2 0

## Norma ⇒ puntual

$({g}_{n}-g)\to 0$

$\lim_{n\to }n\to$ i m 1 g n i g i 2 i m 1 n g n i g i 2 0
Ya que cada término es mayor o igual a cero, todos los términos' $m$ ' deben ser cero. Así, $\lim_{n\to }n\to$ g n i g i 2 0 para todo $i$ . Por lo tanto, ${g}_{n}\to g\text{puntual}$

En un espacio de dimensión finita el teorema anterior ya no es cierto. Probaremos esto con contraejemplos mostrados a continuación.

## Puntual ⇒ norma

Dada la siguiente función: ${g}_{n}(t)=\begin{cases}n & \text{if 0< t< \frac{1}{n}}\\ 0 & \text{otherwise}\end{cases}$ Entonces $\lim_{n\to }n\to$ g n t 0 Esto significa que, ${g}_{n}(t)\to g(t)$ donde para todo $t$ $g(t)=0$ .

Ahora,

$({g}_{n})^{2}=\int \,d t$ g n t 2 t 1 n 0 n 2 n
Ya que la norma de la función se eleva, no puede converger a cualquier función con norma finita.

## Norma ⇒ puntual

Dada la siguiente función: ${g}_{n}(t)=\begin{cases}1 & \text{if 0< t< \frac{1}{n}}\\ 0 & \text{otherwise}\end{cases}\text{si n es par}$ ${g}_{n}(t)=\begin{cases}-1 & \text{if 0< t< \frac{1}{n}}\\ 0 & \text{otherwise}\end{cases}\text{si n es impar}$ Entonces, $({g}_{n}-g)=\int_{0}^{\frac{1}{n}} 1()\,d t=\frac{1}{n}\to 0$ donde $g(t)=0$ para todo $t$ . Entonces, ${g}_{n}\to g\text{en la norma}$ Sin embargo, en $t=0$ , ${g}_{n}(t)$ oscila entre -1 y 1, Y por lo tanto es no convergente. Así, ${g}_{n}(t)$ no tiene convergencia puntual.

## Problemas

Pruebe si las siguientes secuencias tienen convergencia puntual, norma de convergencia, o ambas se mantienen en sus limites.

• ${g}_{n}(t)=\begin{cases}\frac{1}{nt} & \text{if 0< t}\\ 0() & \text{if t\le 0}\end{cases}$
• ${g}_{n}(t)=\begin{cases}e^{-(nt)} & \text{if t\ge 0}\\ 0() & \text{if t< 0}\end{cases}$

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