6. Banach Spaces

Question 1

What does it mean for (X, ∥ · ∥) to be a Banach space?

Answer 1

A Banach space is a complete normed space where every Cauchy Sequence convergences to a limit in X.

Question 2

What does it mean to say that (xn)n≥1 is a Cauchy sequence?

Answer 2

Definition 6.1: the sequence is a Cauchy sequence if for every ε > 0 there exists N ≥ 1 such that ‖xm − xn‖ < ε for all m, n ≥ N .

Question 3

What does it mean to say that (xn)n≥1 converges to x in X?

Answer 3

The sequence converges to x if for every ε > 0 there exists N ≥ 1 such that ‖x − xn‖ < ε for all n ≥ N

Question 4

When describing sequences, what can you say about uniqueness of limits?

Answer 4

Limits are unique, in the sense that every sequence converges to at most one limit.