Functional Analysis 

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1.Theorem: Nested Closed Subset Theorem 

2.Banach Spaces and Banach Algebra 

3.Hilbert Spaces 

4.Linear Operators and Linear Functionals 

5.Non-linear Operator 

6.Lebesque Measure and Lebesque Integral 

7.Simple Connectivity 

8.The $C(X)$ Algebra 

9.Compact Sets 

10.Definition: Sigma-Algebra 

11.Definition: Ring of Sets (measure-theoretic definition) 

12.Definition: Functional