Chapter 6 - Definitions

Question 1

What is the definition for pointwise convergence?

Answer 1

A sequence of functions fsubscriptn: D goes to the real numbers converges pointwise to a function f: D goes to the real numbers if, for each fixed x that exists in D, the real sequence (fsubscriptn(x)) converges to the real number f(x).

Question 2

What is the definition of sup norm?

Answer 2

The sup norm of a bounded function f: D goes to the real numbers is ||f||:= sup{|f(x)|: x exists in D}

Question 3

What is the definition for uniform convergence?

Answer 3

A sequence of bounded function fsubscriptn: D goes to the real numbers converges uniformly to a function f:D goes to the real numbers if the real sequence || fsubscriptn - f|| goes to 0.

Question 4

What is the definition for uniformly Cauchy?

Answer 4

A sequence of bounded functions fsubscriptn: D goes to the real numbers is uniformly Cauchy if, for each epsilon>0, there exists N that exists in the positive integers such that, for all n,m >or= N, ||fsubscriptn - fsubscriptm||< epsilon.