Four Pillars of Functional Analysis

Question

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.

That they are important will be obvious if you continue to read the book. — Dietrich Burde, Apr 01 '18 at 09:19
These theorems are used again and again in functional analysis. — Angina Seng, Apr 01 '18 at 09:27
ya, I know that but is there anything more precise than that @LordSharktheUnknown — Devendra Singh Rana, Apr 01 '18 at 09:28
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. — , Apr 01 '18 at 09:29
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. — littleO, Apr 01 '18 at 09:39
Ok, here's a ig picture overview of how the closed graph theorem is useful. The theorem says that if $T$ has a closed graph then $T$ is bounded. The big picture of how this is useful is that when you can show $T$ has a closed graph you can conclude that $T$ is bounded. (Is that a useful comment? No. Is it a silly comment? Yes. But it is the big picture. If you want to actually understand why these theorems are important, read the book.) — David C. Ullrich, Apr 01 '18 at 16:17

score 5 · Accepted Answer · answered Jul 15 '18 at 06:42

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.

score 0 · Answer 2 · answered Sep 10 '20 at 12:00

0

This article might be of help.

https://www.imsc.res.in/~kesh/trinity.pdf

answered Sep 10 '20 at 12:00

Zxcvasdf

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Four Pillars of Functional Analysis

2 Answers2