# Four Pillars of Functional Analysis

I have come across to a statement in many Functional Analysis books saying that

"Hahn Banach theorem, Uniform Boundedness Principle, Open mapping theorem and Closed graph theorem are the four pillars of Functional Analysis"

I don't exactly know why they are so important, maybe these are used in many parts of Functional Analysis further. can anyone help me, please? thanks and regards in advance.

• What's the question? Apr 1, 2018 at 9:19
• That they are important will be obvious if you continue to read the book. Apr 1, 2018 at 9:19
• These theorems are used again and again in functional analysis. Apr 1, 2018 at 9:27
• Just like ordinary linear algebra has its own toolkit of three-four theorems, linear algebra in Banach spaces has these four pivotal results that capture essential properties of of how continuity of linear functionals works.
– user228113
Apr 1, 2018 at 9:29
• The question is to give a big picture overview of how these theorems are useful in functional analysis. I'm interested to hear the answers as well. Apr 1, 2018 at 9:39

Based on my study of the subject I think I have enough information that I can answer my own question.

Hahn-Banach Theorem: It is so much important because it provides us with the linear functionals to work on various spaces as Functional Analysis is all about the study of functionals.

Open Mapping Theorem: It provides us with the open sets in the topology of the range of the mapping.

Uniform Boundedness Principle: An application of Baire Category theorem. It is further used many times as the uniformity is an important property.

Closed Graph Theorem: Closeness of the graph of a map is enough to prove its boundedness or continuity. This fact is further used many times.