Skip to content Skip to navigation

OpenStax-CNX

You are here: Home » Content » Convergence of Vectors

Navigation

Lenses

What is a lens?

Definition of a lens

Lenses

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

What is in a lens?

Lens makers point to 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 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 ...

Affiliated with (What does "Affiliated with" mean?)

This content is either by members of the organizations listed or about topics related to the organizations listed. Click each link to see a list of all content affiliated with the organization.
  • OrangeGrove display tagshide tags

    This module is included inLens: Florida Orange Grove Textbooks
    By: Florida Orange GroveAs a part of collection: "Signals and Systems"

    Click the "OrangeGrove" link to see all content affiliated with them.

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

  • Rice Digital Scholarship display tagshide tags

    This module is included in aLens by: Digital Scholarship at Rice UniversityAs a part of collection: "Signals and Systems"

    Click the "Rice Digital Scholarship" link to see all content affiliated with them.

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

Also in these lenses

  • Lens for Engineering

    This module is included inLens: Lens for Engineering
    By: Sidney Burrus

    Click the "Lens for Engineering" link to see all content selected in this lens.

  • 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. E-mail the author

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

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 = gn 1 gn 2=1+1n21n gn gn 1 gn 2 1 1 n 2 1 n First we find the following limits for our two gn gn's: limit   n gn 1=1 n gn 1 1 limit   n gn 2=2 n gn 2 2 Therefore we have the following, limit   n gn =g n gn g pointwise, where g=12 g 1 2 .

Example 2

t ,tR: gn t=tn t t gn t t n As done above, we first want to examine the limit limit   n gn t0=limit   n t0n=0 n gn t0 n t0 n 0 where t0R t0 . Thus limit   n gn=g n gn g pointwise where gt=0 g t 0 for all tR t .

Norm Convergence

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

Example 3

g n =1+1n21n g n 1 1 n 2 1 n Let g=12 g 1 2

g n g=1+1n12+21n2=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 limit   n g n g=0 n g n g 0 Therefore, g n g g n g in norm.

Example 4

g n t={tn  if  0t10  otherwise   g n t t n 0 t 1 0 Let gt=0 g t 0 for all tt.

g n tgt=01t2n2d t =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 limit   n g n tgt=0 n g n t g t 0 Therefore, g n tgt 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 igi g n i g i Assuming the above, then g n g2= i =1m g n igi2 g n g 2 i m 1 g n i g i 2 Thus,

limit   n g n g2=limit   n i =1m2= i =1mlimit   n 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

g n g0 g n g 0

limit   n i =1m2= i =1mlimit   n 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, limit   n 2=0 n g n i g i 2 0 forall ii. Therefore, g n 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={n  if  0<t<1n0  otherwise   g n t n 0 t 1 n 0 Then limit   n g n t=0 n g n t 0 This means that, g n tgt g n t g t pointwise where for all tt gt=0 g t 0 .

Now,

g n 2=| g n t|2d t =01nn2d t =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={1  if  0<t<1n0  otherwise   if n is even g n t 1 0 t 1 n 0 if n is even g n t={-1  if  0<t<1n0  otherwise   if n is odd g n t -1 0 t 1 n 0 if n is odd Then, g n g=01n1d t =1n0 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.

Content actions

Download module as:

Add module to:

My Favorites (?)

'My Favorites' is a special kind of lens which you can use to bookmark modules and collections. 'My Favorites' can only be seen by you, and collections saved in 'My Favorites' can remember the last module you were on. You need an account to use 'My Favorites'.

| A lens I own (?)

Definition of a lens

Lenses

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

What is in a lens?

Lens makers point to 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 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