300 CHAPTER 12. BANACH SPACES

12.1.3 Open Mapping TheoremAnother remarkable theorem which depends on the Baire category theorem is the openmapping theorem. Unlike Theorem 12.1.7 it requires both X and Y to be Banach spaces.

Theorem 12.1.8 Let X and Y be Banach spaces, let L ∈L (X ,Y ), and suppose Lis onto. Then L maps open sets onto open sets.

To aid in the proof, here is a lemma.

Lemma 12.1.9 Let a and b be positive constants and suppose

B(0,a)⊆ L(B(0,b)).

ThenL(B(0,b))⊆ L(B(0,2b)).

Proof of Lemma 12.1.9: Let y ∈ L(B(0,b)). There exists x1 ∈ B(0,b) such that

∥y−Lx1∥<a2.

Now this implies2y−2Lx1 ∈ B(0,a)⊆ L(B(0,b)).

Therefore, there exists x2 ∈ B(0,b) such that ∥2y−2Lx1−Lx2∥< a/2. Hence

∥4y−4Lx1−2Lx2∥< a

and there exists x3 ∈ B(0,b) such that

∥4y−4Lx1−2Lx2−Lx3∥< a/2

Continuing in this way, there exist x1,x2,x3,x4, ... in B(0,b) such that∥∥∥∥∥2ny−n

∑i=1

2n−(i−1)L(xi)

∥∥∥∥∥< a

which implies ∥∥∥∥∥y−n

∑i=1

2−(i−1)L(xi)

∥∥∥∥∥=∥∥∥∥∥y−L

(n

∑i=1

2−(i−1)(xi)

)∥∥∥∥∥< 2−na (12.1)

Now consider the partial sums of the series, ∑∞i=1 2−(i−1)xi.∥∥∥∥∥ n

∑i=m

2−(i−1)xi

∥∥∥∥∥≤ b∞

∑i=m

2−(i−1) = b 2−m+2.

Therefore, these partial sums form a Cauchy sequence and so since X is complete, thereexists x = ∑

∞i=1 2−(i−1)xi. Letting n→ ∞ in 12.1 yields ∥y−Lx∥= 0. Now

∥x∥= limn→∞

∥∥∥∥∥ n

∑i=1

2−(i−1)xi

∥∥∥∥∥