Foreword

It's been too long - a month and a half since my last review, and about three months since Analysis I. I've been immersed in my work for CHAI, but reality doesn't grade on a curve, and I want more mathematical firepower.

On the other hand, I've been cooking up something really special, so watch this space!

Analysis II

12: Metric Spaces

Metric spaces; completeness and compactness.

Proving Completeness

It sucks, and I hate it.

13: Continuous Functions on Metric Spaces

Generalized continuity, and how it interacts with the considerations introduced in the previous chapter. Also, a terrible introduction to topology.

There's a lot I wanted to say here about topology, but I don't think my understanding is good enough to break things down - I'll have to read an actual book on the subject.

14: Uniform Convergence

Pointwise and uniform convergence, the Weierstrass $M$ -test, and uniform approximation by polynomials.

Breaking Point

Suppose we have some sequence of functions $f^{(n)} : [0, 1] \to R$ , $f^{(n)} (x) := x^{n}$ , which converge pointwise to the 1-indicator function $f : [0, 1] \to R$ (i.e., $f (1) = 1$ and $0$ otherwise). Clearly, each $f^{(n)}$ is (infinitely) differentiable; however, the limiting function $f$ isn't differentiable at all! Basically, pointwise convergence isn't at all strong enough to stop the limit from "snapping" the continuity of its constituent functions.

Progress

As in previous posts, I mark my progression by sharing a result derived without outside help.

Already proven: $\int_{- 1}^{1} (1 - x^{2})^{N} d x \geq \frac{1}{\sqrt{N}}$ .

Definition. Let $ϵ > 0$ and $0 < δ < 1$ . A function $f : R \to R$ is said to be an $(ϵ, δ)$ -approximation to the identity if it obeys the following three properties:

$f$ is compactly supported on $[- 1, 1]$ .
$f$ is continuous, and $\int_{- \infty}^{\infty} f = 1$ .
$| f (x) | \leq ϵ$ for all $δ \leq | x | \leq 1$ .

Lemma: For every $ϵ > 0$ and $0 < δ < 1$ , there exists an $(ϵ, δ)$ -approximation to the identity which is a polynomial $P$ on $[- 1, 1]$ .

Proof of Exercise 14.8.2(c). Suppose $c \in R, N \in N$ ; define $f (x) := c (1 - x^{2})^{N}$ for $x \in [- 1, 1]$ and $0$ otherwise. Clearly, $f$ is compactly supported on $[- 1, 1]$ and is continuous. We want to find $c, N$ such that the second and third properties are satisfied. Since $(1 - x^{2})^{N}$ is non-negative on $[- 1, 1]$ , $c$ must be positive, as $f$ must integrate to $1$ . Therefore, $f$ is non-negative.

We want to show that $| c (1 - x^{2})^{N} | \leq ϵ$ for all $δ \leq | x | \leq 1$ . Since $f$ is non-negative, we may simplify to $(1 - x^{2})^{N} \leq \frac{ϵ}{c}$ . Since the left-hand... ,