Real Analysis

Definition 1

The extended real line is the set

Definition 2

Definition 3

Norms

Definition 4

Definition 5

Definition 6

The induced norm can be thought of as the maximum gain of the operator.

Definition 7

Sets

Definition 8

Definition 9

Open sets have a boundary which is not included in the set. By convention, we say that the empty set is open.

The opposite of an open set is a closed set.

Definition 10

Closed sets have a boundary which is included in the set.

Convergence

Definition 11

Convergence means that we can always find a finite time such that after that time, all points in the sequence stay within a specified norm ball.

Definition 12

A Cauchy sequence has a looser type of convergence than a convergent sequence since it only requires all elements to in the sequence to be part of the same norm ball after some time instead of requiring the sequence to get closer and closer to a single point.

Theorem 1

Definition 13

Because a complete space requires that Cauchy sequences converge, all cauchy sequences are convergent in a complete space. Two important complete spaces are

  1. Every finite dimensional vector space

A complete normed space is also called a Banach Space.

Contractions

Definition 14

Definition 15

Informally, a contraction is a function which makes distances smaller. Suppose we look at a sequence defined by iterates of a function

Theorem 2 (Contraction Mapping Theorem)

The contraction mapping theorem proves that contractions have a unique fixed points, and that repeatedly applying the contraction will converge to the fixed point.

Continuity

Definition 16

\label{thm:continuity}

We can make the definition of continuity more restrictive by restraining the rate of growth of the function.

Definition 17

Theorem 3

This captures the idea of growing slower than linear in high dimensional space.

Definition 18

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