## 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

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