Convergence of Vectors 2.1 2002/10/11 2002/10/11 Michael Haag mjhaag@rice.edu Michael Haag mjhaag@rice.edu Richard G. Baraniuk richb@rice.edu convergence norm norm convergence point wise pointwise pointwise convergence vector vectors 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 n 1 gn converges pointwise to the limit g if each element of gn converges to the corresponding element in g . Below are few examples to try and help illustrate this idea. gn gn 1 gn 2 1 1 n 2 1 n First we find the following limits for our two gn 's: n gn 1 1 n gn 2 2 Therefore we have the following, n gn g pointwise, where g 1 2 . t t gn t t n As done above, we first want to examine the limit n gn t0 n t0 n 0 where t0 . Thus n gn g pointwise where g t 0 for all t .

Norm Convergence The sequence n 1 gn converges to g in norm if n gn g 0 . Here ˙ is the norm of the corresponding vector space of g n 's. Intuitively this means the distance between vectors g n and g decreases to 0. g n 1 1 n 2 1 n Let g 1 2 g n g 1 1 n 1 2 2 1 n 1 2 1 n 2 1 n 2 2 n Thus n g n g 0 Therefore, g n g in norm. g n t t n 0 t 1 0 Let g t 0 for all 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 Thus n g n t g t 0 Therefore, g n t g t in norm.

Pointwise vs. Norm Convergence For m , pointwise and norm convergence are equivalent. Pointwise ⇒ Norm g n i g i Assuming the above, then g n g 2 i m 1 g n i g i 2 Thus, 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 Norm ⇒ Pointwise 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 Since each term is greater than or equal zero, all 'm' terms must be zero. Thus, n g n i g i 2 0 forall i. Therefore, g n g pointwise In infinite dimensional spaces the above theorem is no longer true. We prove this with counter examples shown below.

Counter Examples Pointwise ⇏ Norm We are given the following function: g n t n 0 t 1 n 0 Then n g n t 0 This means that, g n t g t pointwise where for all t g t 0 . Now, g n 2 t g n t 2 t 1 n 0 n 2 n Since the function norms blow up, they cannot converge to any function with finite norm. Norm ⇏ Pointwise We are given the following function: g n t 1 0 t 1 n 0 if n is even g n t -1 0 t 1 n 0 if n is odd Then, g n g t 1 n 0 1 1 n 0 where g t 0 for all t. Therefore, g n g in norm However, at t 0 , g n t oscillates between -1 and 1, and so it does not converge. Thus, 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. g n t 1 n t 0 t 0 t 0 g n t n t t 0 0 t 0