# Real Analysis

Supremum is essentially the “least upper bound” in a set. It always exists, and is called

$\sup S$

. The opposite of supremum is the infinimum.The infinimum is the “greatest upper bound”. Like the supremum, it always exists, and it is denoted

$\inf S$

. Supremum and Infinimum can be applied to scalar function $f: S\to \mathbb{R}$

by letting

$\sup_{x\in S} f(x) = \sup \{f(x) | x\in S \}.$

Let

$V$

be a vector space of $\mathbb{R}$

, then $\|\cdot\|: V \to \mathbb{R}$

is a norm if $\forall \boldsymbol{x},\boldsymbol{y}\in V, \alpha \in \mathbb{R}$

,

$\|\boldsymbol{x}\| \geq 0, \qquad \boldsymbol{x} = 0 \Leftrightarrow \|\boldsymbol{x}\| = 0, \qquad \|\alpha \boldsymbol{x}\| = |\alpha|\|\boldsymbol{x}\|, \qquad \|\boldsymbol{x} + \boldsymbol{y}\| \leq \|\boldsymbol{x}\| + \|\boldsymbol{y}\|.$

If we have an operator

$A$

which takes vectors from normed space $(X, \|\cdot\|_X)$

and outputs vectors in normed space $(Y, \|\cdot\|_Y)$

, then we can define another norm on the vector space of operators from $X\to Y$

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

If

$V$

is a finite dimensional vector space if and only if all norms of $V$

are equivalent.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.

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

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.

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.

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
- 2.$(C[a,b], \|\cdot\|_\infty)$, the set of continuously differentiable functions on the closed interval$[a,b]$equipped with the infinity norm.

A complete normed space is also called a

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

$\boldsymbol{x}_{k+1} = P(\boldsymbol{x}_k)$

where

$P$

is a function $P:X\to X$

. When does this sequence converge, and to what point will it converge?If

$P:X\to X$

is a contraction on the Banach space $(X, \|\cdot\|)$

, then there is a unique $\boldsymbol{x}^*\in X$

such that $P(\boldsymbol{x}^*) = \boldsymbol{x}^*$

and $\forall \boldsymbol{x}_0\in X$

, the sequence $\boldsymbol{x}_{n+1} = P(\boldsymbol{x}_n)$

converges to $\boldsymbol{x}^*$

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

A function

$h:V\to W$

on normed spaces $(V, \|\cdot\|_V)$

and $(W, \|\cdot\|_W)$

is continuous at a point $\boldsymbol{x}_0$

if $\forall \epsilon > 0, \exists \delta > 0$

such that

$\|\boldsymbol{x}-\boldsymbol{x}_0\|_V < \delta \implies \|h(\boldsymbol{x}) - h(\boldsymbol{x_0})\|_W < \epsilon$

\label{thm:continuity}

Continuity essentially means that given an

$\epsilon-$

ball in $W$

, we can find a $\delta-$

ball in $V$

which is mapped to the ball in $W$

. If a function is continuous at all points $\boldsymbol{x}_0$

, then we say the function is continuous.We can make the definition of continuity more restrictive by restraining the rate of growth of the function.

A function

$h:V\to W$

on normed spaces $(V, \|\cdot\|_V)$

and $(W, \|\cdot\|_W)$

is Lipschitz continuous at $\boldsymbol{x}_0\in V$

if $\exists r > 0$

and $L < \infty$

such that

$\forall \boldsymbol{x}, \boldsymbol{y}\in B_r(\boldsymbol{x}_0),\ \|h(\boldsymbol{x}) - h(\boldsymbol{y})\|_W \leq L \|\boldsymbol{x} - \boldsymbol{y}\|_V$

A good interpretation of Lipschitz Continuity is that given two points in a ball around

$\boldsymbol{x}_0$

, the slope of the line connecting those two points is less than $L$

. It means that the function is growing slower than linear for some region around $\boldsymbol{x}_0$

. Lipschitz continuity implies continuity. If a function is lipschitz continuous with respect to one norm, it is also lipschitz continuous with respect to all equivalent norms.When the function

$h$

is a function on $\mathbb{R}^n$

and is also differentiable, then Lipschitz continuity is easy to determine.For a differentiable function

$h:\mathbb{R}^n\to\mathbb{R}^n$

,

$\exists r>0, L < \infty, \boldsymbol{x}_0\in\mathbb{R}^n,\ \forall \boldsymbol{x}\in B_r(\boldsymbol{x}_0), \left\lvert\left\lvert\frac{\partial h}{\partial \boldsymbol{x}}\right\rvert\right\rvert_2 \leq L$

implies Lipschitz Continuity at

$\boldsymbol{x}_0$

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

Last modified 1yr ago