Pointwise and Uniform Convergence

Definition: Letbe a subset ofand letbe a sequence of functions defined onWe say thatconverges pointwise onif exists for each point

This means thatis a real number that depends only on

Ifis pointwise convergent then the function defined by

for everyis called the pointwise limit of the sequence

More formally, given anyand there exists a natural numberdepending only onandsuch that

for every

Definition Letbe a subset ofand letbe a sequence of real valued functions defined onThenconverges uniformly toif given anythere exists a natural numbersuch thatfor everydepends only on so uniform convergence implies pointwise convergence., but the converse is not true.


Asfor eachso converges pointwise to zero.

Suppose now thatandthenandsois not uniformly convergent.

Add comment

Security code