1.2 Important Definitions

Question 1

What is an open ball/sphere?

Answer 1

An open ball with center x₀ and radius r > 0 is the set B(x₀, r) = { x belonging to X : ||x₀ - x|| < r } where X is a normed linear space.

Question 2

What is a closed ball/sphere?

Answer 2

A closed ball/sphere with center x₀ and radius r > 0 is the set B(x₀, r) = {x belonging to X : ||x - x₀|| <= r} where X is a normed linear space.

Question 3

What is the convergence of a sequence?

Answer 3

Let X be a normed linear space and {xₙ} be a sequence in X. The sequence {xₙ} of elements of X converges to a point x belonging to X if for every epsilon > 0 there exists a positive integer N such that: ||xₙ - x|| < epsilon, for all n >= N.

Alternatively, {xₙ} converges to the point x in X if and only if: lim ||xₙ - x|| = 0 as n -> infinity.

Question 4

What is a Cauchy sequence?

Answer 4

Let {xₙ} be a sequence in the normed linear space X. {xₙ} is a Cauchy sequence if for every epsilon > 0 there exists a positive integer N such that: ||xₘ - xₙ|| < epsilon, for all n, m >= N.

Alternatively, {xₙ} is a Cauchy sequence if and only if: lim ||xₘ - xₙ|| = 0 as m, n -> infinity.