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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.

Note: You are viewing an old version of this document. The latest version is available here.

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+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: 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 ,t∈R: 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 t0∈R t0 . Thus limit   n → ∞ gn=g n gn g pointwise where g⁢t=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 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+1n2−1n g n 1 1 n 2 1 n Let g=12 g 1 2

∥ g n −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 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  0≤t≤10  otherwise   g n t t n 0 t 1 0 Let g⁢t=0 g t 0 for all tt.

∥ g n ⁢t−g⁢t∥=∫01t2n2d t =t33⁢n2| n =01=13⁢n2 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 ⁢t−g⁢t∥=0 n g n t g t 0 Therefore, g n ⁢t→g⁢t 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→g⁢i g n i g i Assuming the above, then ∥ g n −g∥2=∑ i =1m g n ⁢i−g⁢i2 g n g 2 i m 1 g n i g i 2 Thus,

limit   n → ∞ ∥ g n −g∥2=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 −g∥→0 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 ⁢t→g⁢t g n t g t pointwise where for all tt g⁢t=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 =1n→0 g n g t 1 n 0 1 1 n 0 where g⁢t=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={1n⁢t  if  0<t0  if  t≤0 g n t 1 n t 0 t 0 t 0
  2. g n ⁢t={e−(n⁢t)  if  t≥00  if  t<0 g n t n t t 0 0 t 0

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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?

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