458 CHAPTER 17. BANACH SPACES

where ∑k |dk|2 = 1 in contradiction to the linear independence of the {x1, · · · ,xn}. Hence itmust be the case that cp is bounded in Fn. Then taking a subsequence, still denoted as p, itcan be assumed cp→ c and then in 17.4.9 it follows

x =n

∑k=1

ckxk ∈ span(x1, · · · ,xn) .

Proposition 17.4.2 Let E be a separable Banach space. Then there exists an increasingsequence of subspaces, {Fn} such that dim(Fn+1)−dim(Fn) ≤ 1 and equals 1 for all n ifthe dimension of E is infinite. Also ∪∞

n=1Fn is dense in E. In the case where E is infinitedimensional, Fn = span(e1, · · · ,en) where for each n

dist(en+1,Fn)≥12

(17.4.10)

and defining,Gk ≡ span

({e j : j ̸= k

})dist(ek,Gk)≥

14

. (17.4.11)

Proof: Since E is separable, so is ∂B(0,1) , the boundary of the unit ball. Let {wk}∞

k=1be a countable dense subset of ∂B(0,1).

Let e1 = w1. Let F1 = Fe1. Suppose Fn has been obtained and equals span(e1, · · · ,en)where {e1, · · · ,en} is independent, ||ek||= 1, and

dist(en,span(e1, · · · ,en−1))≥12.

For each n, Fn is closed by Theorem 17.4.1.If Fn contains {wk}∞

k=1 , let Fm = Fn for all m > n. Otherwise, pick w ∈ {wk} to be thepoint of {wk}∞

k=1 having the smallest subscript which is not contained in Fn. Then w is at apositive distance, λ from Fn because Fn is closed. Therefore, there exists y ∈ Fn such thatλ ≤ ||y−w|| ≤ 2λ . Let en+1 =

w−y||w−y|| . It follows

w = ||w− y||en+1 + y ∈ span(e1, · · · ,en+1)≡ Fn+1

Then if x ∈ span(e1, · · · ,en) ,

||en+1− x|| =

∣∣∣∣∣∣∣∣ w− y||w− y||

− x∣∣∣∣∣∣∣∣

=

∣∣∣∣∣∣∣∣ w− y||w− y||

− ||w− y||x||w− y||

∣∣∣∣∣∣∣∣≥ 1

2λ||w− y−||w− y||x||

≥ λ

2λ=

12.