2.2 Theorems of Banach Spaces

Question 1

List out the four theorems of Banach spaces.

Answer 1

Acrostic: Tity On Camera
1. The Banach fixed point theorem
2. The Hahn-Banach theorem
3. Open Mapping theorem
4. Closed graph theorem

Question 2

What is the Banach fixed point theorem?

Answer 2

Every contraction mapping on a non-empty complete metric space has a unique fixed point

Question 3

What is the Hahn-Banach theorem?

Answer 3

Every linear function on a subspace of a Banach space can be extended to a linear function on the whole space with the same norm

Question 4

What is the Open mapping theorem?

Answer 4

Every surjective continuous linear operator between Banach spaces is an open map

Question 5

What is the closed graph theorem?

Answer 5

A linear operator between Banach spaces is continuous if its graph is closed