554 CHAPTER 21. BANACH SPACES

the 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 as desired. 21.11 follows from the

construction. It remains to verify 21.12.Let y∈Gk. Thus for some n,y=∑

k−1j=1 c je j+∑

nj=k+1 c je j and I need to show ∥y− ek∥≥

1/4. Without loss of generality, cn ̸= 0 and n > k. Suppose 21.12 does not hold for somesuch y so that ∥∥∥∥∥ek−

(k−1

∑j=1

c je j +n

∑j=k+1

c je j

)∥∥∥∥∥< 14. (21.13)

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 21.13. By the construction

14>

∥∥∥∥∥∥∥ek−cnen︷ ︸︸ ︷

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. ■

Definition 21.4.3 A Banach space X has a Schauder basis {ek}∞

k=1 if for everyx ∈ X , there are unique scalars ck such that x = ∑

∞k=1 ckxk. This is different than a basis

because you allow countable sums. For example, you might consider Fourier series.

21.5 Weak And Weak ∗ TopologiesProposition 21.4.2 shows that in infinite dimensional space, closed and bounded will notbe compact. However, in applications one would like to be able to get convergence ofsubsequences. This involves asking for less than norm convergence and the concept ofweak topologies.

21.5.1 Basic Definitions

Let X be a Banach space and let X ′ be its dual space.1 For A′ a finite subset of X ′, denoteby ρA′ the function defined on X

ρA′ (x)≡ maxx∗∈A′|x∗ (x)| (21.14)

1Actually, all this works in much more general settings than this.