17.1. THEOREMS BASED ON BAIRE CATEGORY 439

uniformly. As an example of a situation in which pointwise bounded does not imply uni-formly bounded, consider the functions fα (x)≡X(α,1) (x)x−1 for α ∈ (0,1). Clearly eachfunction is bounded and the collection of functions is bounded at each point of (0,1), butthere is no bound for all these functions taken together. One problem is that (0,1) is not aBanach space. Therefore, the functions cannot be linear.

Theorem 17.1.8 Let X be a Banach space and let Y be a normed linear space. Let{Lα}α∈Λ be a collection of elements of L (X ,Y ). Then one of the following happens.

a.) sup{||Lα || : α ∈ Λ}< ∞

b.) There exists a dense Gδ set, D, such that for all x ∈ D,

sup{||Lα x|| α ∈ Λ}= ∞.

Proof: For each n ∈ N, define

Un = {x ∈ X : sup{||Lα x|| : α ∈ Λ}> n}.

Then Un is an open set because if x ∈Un, then there exists α ∈ Λ such that

||Lα x||> n

But then, since Lα is continuous, this situation persists for all y sufficiently close to x, sayfor all y ∈ B(x,δ ). Then B(x,δ )⊆Un which shows Un is open.

Case b.) is obtained from Theorem 17.1.2 if each Un is dense.The other case is that for some n, Un is not dense. If this occurs, there exists x0 and r > 0

such that for all x ∈ B(x0,r), ||Lα x|| ≤ n for all α . Now if y ∈ B(0,r), x0 + y ∈ B(x0,r).Consequently, for all such y, ||Lα(x0+y)|| ≤ n. This implies that for all α ∈Λ and ||y||< r,

||Lα y|| ≤ n+ ||Lα(x0)|| ≤ 2n.

Therefore, if ||y|| ≤ 1,∣∣∣∣ r

2 y∣∣∣∣< r and so for all α ,

||Lα

( r2

y)|| ≤ 2n.

Now multiplying by r/2 it follows that whenever ||y|| ≤ 1, ||Lα (y)|| ≤ 4n/r. Hence casea.) holds.

17.1.3 Open Mapping TheoremAnother remarkable theorem which depends on the Baire category theorem is the openmapping theorem. Unlike Theorem 17.1.8 it requires both X and Y to be Banach spaces.

Theorem 17.1.9 Let X and Y be Banach spaces, let L ∈L (X ,Y ), and suppose L is onto.Then L maps open sets onto open sets.

To aid in the proof, here is a lemma.