Chapter 12

Banach Spaces12.1 Theorems Based on Baire Category

Banach spaces are complete normed linear spaces. These have been mentioned throughoutthe book so far. In this chapter are the most significant theorems relative to Banach spacesand more generally normed linear spaces. The main theorems to be presented here are theuniform boundedness theorem, the open mapping theorem, the closed graph theorem, andthe Hahn Banach Theorem. The first three of these theorems come from the Baire categorytheorem which is about to be presented. They are topological in nature. The Hahn Banachtheorem has nothing to do with topology. First some definitions and a review of the notionof a Banach space. Always we are considering a complete normed linear space of the sortdiscussed earlier. As noted earlier, Banach spaces are all examples of metric space so allthe theory of metric space applies. In particular,

Proposition 12.1.1 The open ball B(z,r) is an open set. Also B(z,r) = D(z,r) ≡{x:∥x− z∥ ≤ r} . In an arbitrary metric space, B(z,r)⊆ D(z,r).

Proof: First note that D(z,r) is closed because x→ ∥x− z∥ is continuous by Theorem2.4.8 for example, so inverse images of closed sets are closed. Thus D(z,r) is a closed setcontaining B(z,r) so B(z,r) ⊆ D(z,r). In normed linear space these are equal because if∥z− x∥ = r, then for n ∈ N, consider xn = x+ n−1

n (z− x). ∥xn− x∥ = n−1n r < r so xn ∈

B(z,r) but ∥xn− z∥=∥∥x+ n−1

n (z− x)− z∥∥= 1

n ∥x− z∥ so z, being the limit of a sequenceof points of B(z,r) is in B(z,r). ■

Note here that equality does not work in general metric space because you could havean infinite set with the metric d (x,y) = 0 if x = y and 1 if x ̸= y. Then B(x,1) would consistof only x while D(x,1) would yield the whole set.

Recall also that |∥z∥−∥w∥| ≤ ∥z−w∥ , note ∥z∥= ∥z−w+w∥≤ ∥z−w∥+∥w∥whichimplies ∥z∥−∥w∥≤ ∥z−w∥ and now switching z and w, yields ∥w∥−∥z∥≤ ∥z−w∥whichimplies |∥w∥−∥z∥| ≤ ∥w− z∥ . This was done earlier. It is just another version of thetriangle inequality.

Also recall the definition of a Cauchy sequence.

Definition 12.1.2 {xn} is called a Cauchy sequence if for every ε > 0 there existsN such that if m,n≥ N, then ∥xn− xm∥< ε.

As discussed earlier in Section 2.9,

Definition 12.1.3 L (X ,Y ) is the space of continuous linear maps from X to Y .Recall also that if L ∈L (X ,Y ) ,∥L∥ ≡ sup{∥Lx∥ : ∥x∥ ≤ 1} and that this is well definedand ∥L◦M∥ ≤ ∥L∥∥M∥ .

As noted earlier, in Section 2.9, whenever you have L∈L (Rp,Rq) , L is automaticallycontinuous. However, in infinite dimensional settings, this might not hold. Here is a simpleexample.

Example 12.1.4 Let V denote all linear combinations of functions of the form e−αx2for

α > 0. Thus typical elements of V are of the form ∑nk=1 β ke−αkx2

. Let L : V → C be given

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