Four Pillars of Functional Analysis

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.

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Functional Analysis

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