17.4. CLOSED SUBSPACES 459

This has shown the existence of an increasing sequence of subspaces, {Fn} as describedabove. It remains to show the union of these subspaces is dense. First note that the union ofthese subspaces must contain the {wk}∞

k=1 because if wm is missing, then it would contradictthe construction at the mth step. That one should have been chosen. However, {wk}∞

k=1 isdense in ∂B(0,1). If x ∈ E and x ̸= 0, then x

||x|| ∈ ∂B(0,1) then there exists

wm ∈ {wk}∞

k=1 ⊆ ∪∞n=1Fn

such that∣∣∣∣∣∣wm− x

||x||

∣∣∣∣∣∣< ε

||x|| . But then

||||x||wm− x||< ε

and so ||x||wm is a point of ∪∞n=1Fn which is within ε of x. This proves ∪∞

n=1Fn is dense asdesired. 17.4.10 follows from the construction. It remains to verify 17.4.11.

Let y ∈ Gk. Thus for some n,

y =k−1

∑j=1

c je j +n

∑j=k+1

c je j

and I need to show ||y− ek|| ≥ 1/4. Without loss of generality, cn ̸= 0 and n > k. Suppose17.4.11 does not hold for some such y so that∣∣∣∣∣

∣∣∣∣∣ek−

(k−1

∑j=1

c je j +n

∑j=k+1

c je j

)∣∣∣∣∣∣∣∣∣∣< 1

4. (17.4.12)

Then from the construction,

14

> |cn|

∣∣∣∣∣∣∣∣∣∣ek−

(k−1

∑j=1

(c j/cn)e j +n−1

∑j=k+1

(c j/cn)e j + en

)∣∣∣∣∣∣∣∣∣∣

≥ |cn|12

and so |cn|< 1/2. Consider the left side of 17.4.12. By the construction∣∣∣∣∣∣∣∣∣∣cn (ek− en)+(1− cn)ek−

(k−1

∑j=1

c je j +n−1

∑j=k+1

c je j

)∣∣∣∣∣∣∣∣∣∣

≥ |1− cn|− |cn|

∣∣∣∣∣∣∣∣∣∣(ek− en)−

(k−1

∑j=1

(c j/cn)e j +n−1

∑j=k+1

(c j/cn)e j

)∣∣∣∣∣∣∣∣∣∣

≥ |1− cn|− |cn|12≥ 1− 3

2|cn|> 1− 3

212=

14,

a contradiction. This proves the desired estimate.