Skip to content Skip to navigation

Connexions

You are here: Home » Content » Convergence of Vectors

Navigation

Content Actions
  • Download module PDF
  • Add to ...
    Add the module to:
    • My Favorites
    • A lens
    • An external social bookmarking service
    • My Favorites (What is 'My Favorites'?)
      'My Favorites' is a special kind of lens which you can use to bookmark modules and collections directly in Connexions. 'My Favorites' can only be seen by you, and collections saved in 'My Favorites' can remember the last module you were on. You need a Connexions account to use 'My Favorites'.
    • A lens (What is a lens?)

      Definition of a lens

      Lenses

      A lens is a custom view of Connexions content. You can think of it as a fancy kind of list that will let you see Connexions through the eyes of organizations and people you trust.

      What is in a lens?

      Lens makers point to Connexions materials (modules and collections), creating a guide that includes their own comments and descriptive tags about the content.

      Who can create a lens?

      Any individual Connexions member, a community, or a respected organization.

      What are tags? tag icon

      Tags are descriptors added by lens makers to help label content, attaching a vocabulary that is meaningful in the context of the lens.

    • External bookmarks
  • E-mail the author
  • Rate this module (How does the rating system work?)

    Rating system

    Ratings

    Ratings allow you to judge the quality of modules. If other users have ranked the module then its average rating is displayed below. Ratings are calculated on a scale from one star (Poor) to five stars (Excellent).

    How to rate a module

    Hover over the star that corresponds to the rating you wish to assign. Click on the star to add your rating. Your rating should be based on the quality of the content. You must have an account and be logged in to rate content.

    		(0 ratings)

Lenses

What is a lens?

Definition of a lens

Lenses

A lens is a custom view of Connexions content. You can think of it as a fancy kind of list that will let you see Connexions through the eyes of organizations and people you trust.

What is in a lens?

Lens makers point to Connexions materials (modules and collections), creating a guide that includes their own comments and descriptive tags about the content.

Who can create a lens?

Any individual Connexions member, a community, or a respected organization.

What are tags? tag icon

Tags are descriptors added by lens makers to help label content, attaching a vocabulary that is meaningful in the context of the lens.

This content is ...

In these lenses

  • richb's DSP display tagshide tags

    This module is included inLens: richb's DSP resources
    By: Richard BaraniukAs a part of collection:"Signals and Systems"

    Comments:

    "My introduction to signal processing course at Rice University."

    Click the "richb's DSP" link to see all content selected in this lens.

    Click the tag icon tag icon to display tags associated with this content.

Recently Viewed

This feature requires Javascript to be enabled.

Tags

(What is a tag?)

These tags come from the endorsement, affiliation, and other lenses that include this content.

Convergence of Vectors

Module by: Michael Haag

Summary: This modules presents two common types of convergence, pointwise and norm, and discusses their properties, differences, and relationships with one another.

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.

Convergence of Vectors

We now discuss pointwise and norm convergence of vectors. Other types of convergence also exist, and one in particular, uniform convergence, can also be studied. For this discussion , we will assume that the vectors belong to a normed vector space.

Pointwise Convergence

A sequence gn |n=1∞ n 1 gn converges pointwise to the limit g g if each element of gn gn converges to the corresponding element in g g. Below are few examples to try and help illustrate this idea.

Example 1

gn= gn1gn2 = 1+1n2−1n gn gn 1 gn 2 1 1 n 2 1 n First we find the following limits for our two gn gn's: limn→∞gn1=1 n gn 1 1 limn→∞gn2=2 n gn 2 2 Therefore we have the following, limn→∞gn=g n gn g pointwise, where g=12 g 1 2 .

Example 2

∀t,t∈ℝ:gnt=tn t t gn t t n As done above, we first want to examine the limit limn→∞gnt0=limn→∞t0n=0 n gn t0 n t0 n 0 where t0∈ℝ t0 . Thus limn→∞gn=g n gn g pointwise where gt=0 g t 0 for all t∈ℝ t .

Norm Convergence

The sequence gn |n=1∞ n 1 gn converges to gg in norm if limn→∞∥gn−g∥=0 n gn g 0 . Here ∥˙∥ ˙ is the norm of the corresponding vector space of gn g n 's. Intuitively this means the distance between vectors gn g n and g g decreases to 00.

Example 3

gn=1+1n2−1n g n 1 1 n 2 1 n Let g=12 g 1 2

∥gn−g∥=1+1n−12+2−1n2=1n2+1n2=2n g n g 1 1 n 1 2 2 1 n 1 2 1 n 2 1 n 2 2 n (1)
Thus limn→∞∥gn−g∥=0 n g n g 0 Therefore, gn→g g n g in norm.

Example 4

g n t=tnif0≤t≤10otherwise g n t t n 0 t 1 0 Let gt=0 g t 0 for all tt.

∥ g n t−gt∥=∫01t2n2dt=t33n2|n=01=13n2 g n t g t t 1 0 t 2 n 2 n 0 1 t 3 3 n 2 1 3 n 2 (2)
Thus limn→∞∥ g n t−gt∥=0 n g n t g t 0 Therefore, g n t→gt g n t g t in norm.

Pointwise vs. Norm Convergence

Theorem 1

For ℝm m , pointwise and norm convergence are equivalent.

Proof: Pointwise ⇒ Norm

g n i→gi g n i g i Assuming the above, then ∥gn−g∥2=∑i=1m g n i−gi2 g n g 2 i m 1 g n i g i 2 Thus,

limn→∞∥gn−g∥2=limn→∞∑i=1m2=∑i=1mlimn→∞2=0 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 (3)

Proof: Norm ⇒ Pointwise

∥gn−g∥→0 g n g 0

limn→∞∑i=1m2=∑i=1mlimn→∞2=0 n i m 1 g n i g i 2 i m 1 n g n i g i 2 0 (4)
Since each term is greater than or equal zero, all 'mm' terms must be zero. Thus, limn→∞2=0 n g n i g i 2 0 forall ii. Therefore, gn→g pointwise g n g pointwise

note:

In infinite dimensional spaces the above theorem is no longer true. We prove this with counter examples shown below.

Counter Examples

Example 5: Pointwise ⇏ Norm

We are given the following function: g n t=nif0<t<1n0otherwise g n t n 0 t 1 n 0 Then limn→∞ g n t=0 n g n t 0 This means that, g n t→gt g n t g t pointwise where for all tt gt=0 g t 0 .

Now,

∥ g n ∥2=∫-∞∞| g n t|2dt=∫01nn2dt=n→∞ g n 2 t g n t 2 t 1 n 0 n 2 n (5)
Since the function norms blow up, they cannot converge to any function with finite norm.

Example 6: Norm ⇏ Pointwise

We are given the following function: g n t=1if0<t<1n0otherwise if n is even g n t 1 0 t 1 n 0 if n is even g n t=-1if0<t<1n0otherwise if n is odd g n t -1 0 t 1 n 0 if n is odd Then, ∥ g n −g∥=∫01n 1 dt=1n→0 g n g t 1 n 0 1 1 n 0 where gt=0 g t 0 for all tt. Therefore, g n →g in norm g n g in norm However, at t=0 t 0 , g n t g n t oscillates between -1 and 1, and so it does not converge. Thus, g n t g n t does not converge pointwise.

Problems

Prove if the following sequences are pointwise convergent, norm convergent, or both and then state their limits.

  1. g n t=1ntif0<t 0 ift≤0 g n t 1 n t 0 t 0 t 0
  2. g n t=ⅇ-ntift≥0 0 ift<0 g n t n t t 0 0 t 0

Comments, questions, feedback, criticisms?

Send feedback