5. Convergence in Normed Spaces

Question 1

Let X be a normed space. When does a sequence (Xn)n converge to x.

Answer 1

Let X be a normed space and suppose that xn ∈X for n≥1 and x∈X. The sequence (xn)n≥1 is said to converge to (the limit) x if for every ε > 0 there exists N≥1 such that ∥xn−x∥ < ε for all n≥N. We then write xn →x as n→∞,or lim n→∞ xn = x.

Question 2

What can you say about uniqueness of limits in normed spaces

Answer 2

Let (xn)n≥1 be a sequence in a normed space X
and suppose that x, y ∈ X are such that xn → x and xn → y as n → ∞. Then x = y.

Question 3

Let X be a normed space and let Y be a subset of X. When is Y dense in X ?

Answer 3

We say that Y is dense in X if for every x∈X and for every ε>0 there exists y∈Y such that ∥x − y∥ < ε.