<< Chapter < Page Chapter >> Page >
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 .

Got questions? Get instant answers now!

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 .

Got questions? Get instant answers now!

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.

Got questions? Get instant answers now!

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.

Got questions? Get instant answers now!

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.

Got questions? Get instant answers now!

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.

Got questions? Get instant answers now!


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

price elasticity of demand is the degree of responsiveness of a quantity demanded to the change in price of the commodity in question.
Gladys Reply
what is the importance of learning economics?
Thelma Reply
it helps to make the correct choice
it helps firm to produce products that will bring more profit
the difference between needs and wants
londiwe Reply
needs are things that we basically can't live without wants are just luxury things
needs are things without them we can't live but want are things without we can live
what is education
it's a process in which we give or receiving methodical instructions
what is mixed economy
what is a deadweight loss? how monopoly creates a deadweight loss?
Ashraf Reply
who are u?
what it this
hi y'all
how does group chat help y'all 🤔
hi y'all
how does group chat help y'all 🤔
how does group chat help y'all 🤔
to learn from one another
oh okay
what is type of economic
taiwo Reply
how to understand basics of economics
Aarif Reply
what is demand schedle
Princess Reply
When you make a Scedule of the demand you made
this is helpful for rbi grade b
Prema Reply
What is macroeconomics
Kauna Reply
It's one of the two branches of Economics that deal with the aggregate economy.
it's about inflation, occupation, gdp and so on
What is differences between Microeconomics and Macroeconomic?
microeconomics focuses on the action of individual agents in the economy such as businesses, workers and household. while macroeconomics looks at the economy as a whole. it focuses on broad issues in the economy such as government deficit, economy growth, levels of exports and imports, and
inflationary increase in prices
a price floor of 24 imposed
Annie Reply
monopolistic competition
bintu Reply
any one there to answer my question
Richard Reply
Fixed Costs per week Variable Costs per bear Rent & Rates of Factory Hire & machines Heating & Lighting Repayment of Bank Loan K100.00 K45.00 K5.00 K50.00 Materials Foam Wages K6.00 K1.00 K1.00 Total K200.00 K8.00
one of the scarce resources that constrain our behaviour is time. each of us has only 24 hours in a day. how do you go about allocating your time in a given day among completing alternatives? once you choose a most important use of time. why do you not spend all your time to it. use the notion of op
naknak Reply
mohsina mala..Bangla app hobe na
Dipam Reply
mani Baba. First learn the spelling of Economics
Dipam Reply
Economics- The study of how people use their limited resources to tey and satisfy unlimited wants.
etar bangla apps hobe na?
in a comparison of the stages of meiosis to the stage of mitosis, which stages are unique to meiosis and which stages have the same event in botg meiosis and mitosis
Leah Reply
Got questions? Join the online conversation and get instant answers!
Jobilize.com Reply

Get the best Algebra and trigonometry course in your pocket!

Source:  OpenStax, Señales y sistemas. OpenStax CNX. Sep 28, 2006 Download for free at http://cnx.org/content/col10373/1.2
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Señales y sistemas' conversation and receive update notifications?