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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 1 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 g n 1 g n 2 1 1 n 2 1 n Primero encontramos los siguiente limites para nuestras dos g n 's: n g n 1 1 n g n 2 2 Después tenemos el siguiente, n g n g puntual, donde g 1 2 .

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t t g n t t n Como se hizo anteriormente, primero examinamos el límite n g n t 0 n t 0 n 0 donde t 0 . Por lo tanto n g n g puntualmente donde g t 0 para toda t .

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Norma de convergencia

La secuencia n 1 g n converge a g en la norma si n 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 1 1 n 2 1 n Sea g 1 2

g n g 1 1 n 1 2 2 1 n 1 2 1 n 2 1 n 2 2 n
Asi n g n g 0 , Por lo tanto, g n g en la norma.

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g n t t n 0 t 1 0 Sea g t 0 para todo t .

g n t g t t 1 0 t 2 n 2 n 0 1 t 3 3 n 2 1 3 n 2
Asi n g n t g t 0 Por lo tanto, g n t g t en la norma.

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Puntual vs.norma de convergencia

Para m , la convergencia puntual y la norma de convergencia es equivalente.

Puntual ⇒ norma

g n i g i Asumiendo lo anterior, entonces g n g 2 i m 1 g n i g i 2 Así,

n 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 0

n 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, n g n i g i 2 0 para todo i . Por lo tanto, g n g puntual

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

Contra ejemplos

Puntual ⇒ norma

Dada la siguiente función: g n t n 0 t 1 n 0 Entonces n g n t 0 Esto significa que, g n t g t pointwise donde para todo t g t 0 .


g n 2 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.

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Norma ⇒ puntual

Dada la siguiente función: g n t 1 0 t 1 n 0 si n es par g n t -1 0 t 1 n 0 si n es impar Entonces, g n g t 1 n 0 1 1 n 0 donde g t 0 para todo t . Entonces, g n g 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.

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

  • g n t 1 n t 0 t 0 t 0
  • g n t n t t 0 0 t 0

Questions & Answers

Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
what school?
biomolecules are e building blocks of every organics and inorganic materials.
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
sciencedirect big data base
Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
Damian Reply
absolutely yes
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Akash Reply
it is a goid question and i want to know the answer as well
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Do somebody tell me a best nano engineering book for beginners?
s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
what is the actual application of fullerenes nowadays?
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
what is the Synthesis, properties,and applications of carbon nano chemistry
Abhijith Reply
Mostly, they use nano carbon for electronics and for materials to be strengthened.
is Bucky paper clear?
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
so some one know about replacing silicon atom with phosphorous in semiconductors device?
s. Reply
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Do you know which machine is used to that process?
how to fabricate graphene ink ?
for screen printed electrodes ?
What is lattice structure?
s. Reply
of graphene you mean?
or in general
in general
Graphene has a hexagonal structure
On having this app for quite a bit time, Haven't realised there's a chat room in it.
what is biological synthesis of nanoparticles
Sanket Reply
how did you get the value of 2000N.What calculations are needed to arrive at it
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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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