Articles of analysis

Hausdorff Dimension of a manifold of dimension n?

Let’s say that $M$ is a differentiable manifold of dimension $n$. (This includes that $M$ is nonempty and second countable, so that it can be embedded into some Euclidean space, and is thus metrizable.) Is it well-defined to say that $M$ has Hausdorff dimension $n$ even though it is only a metrizable topological space? I.e. […]

How to reduce this to Sturm-Liouville form?

I have the ODE $$-(1-x^2) \frac{d^2 f(x)}{dx^2} + x \frac{df(x)}{dx}+g(x)f(x)=\lambda f(x)$$ and I want to reduce it to Sturm-liouville form. The problem is that we don’t have $2x$ but just $x$. otherwise it would be similar to the Legendre differential equation. Could anybody help me with that? By the way, does this mean that the […]

Proving that $A \mapsto \sup\{ \mu E \mid A \supset E \in \Sigma, \mu E < \infty\}$ is an inner measure

Let $(X,\Sigma, \mu)$ be a measure space and define $m: 2^X \to [0,\infty]$ by $m A = \sup\{ \mu E \mid A \supset E \in \Sigma, \mu E < \infty\}$. Show that $m$ is an inner measure. There are $4$ axioms to be checked, three of which are pretty easy to do. Hence, I am […]

Lower semi-continuous function which is unbounded on compact set.

Every lower semi-continuous functions attains an infimum/minimum on a compact set, do you know examples of lower semi-continuous functions which are unbounded and/or don’t attain their maximum/supremum?

In a $p$-adic vector space, closest point on (and distance from) a plane to a given point?

Let $\| x \| =\sqrt{x^T x}$ be the Euclidean norm on $\mathbb{R}^n$. Consider the point $z \in \mathbb{R}^n$, and the plane $P = \{x \in \mathbb{R}^n : a^T x = b\}$ where $0 \neq a \in \mathbb{R}^n$, $b \in \mathbb{R}$. Orthogonal projection gives the point \begin{align}\label{1}\tag{1} y = z – \frac{(a^T z – b)}{a^T a} […]

How do we introduce subtraction from these field axioms?

I am familiar with two different sets of field axioms. The first one is from “Mathematical Analysis” by Apostol. It has the first $3$ usual axioms, but the $4^{th}$ one is different: Axiom 1: Commutative Laws $x+y=y+x$, $xy=yx$ Axiom 2: Associative Laws $x+(y+z)=(x+y)+z$, $x(yz)=(xy)z$ Axiom 3: Distributive Law $x(y+z)=xy+xz$ Axiom 4: Given any two real […]

Is this sequence relatively compact?

Let $M$ be a locally compact connected complete metric space with metric $d$. Define $\mathcal{F}$ to be the family of continuous, distance decreasing mappings from $M$ to itself with the property that $f(p)=p$ for some $p\in M$. Now let’s fix $q\in M$ where $q\neq p$ and pick $\{f_n\}$ any sequence in $\mathcal{F}$. My question is: […]

Spivak's Calculus – Chapter 1 Question 1.3

I’m reading through Spivak’s Calculus, and am not sure where to start in proving the following: if $x^2 = y^2$, then $x = y$ or $x = -y$ Particularly given that only the following properties can be used to justify each step of the proof: Any hints on how to start would be much appreciated!

Product of weak/strong converging sequences

Let’s consider two sequences $u_n$ and $v_n$ such that $$u_n\to u\,,\,\,\,\rm{in}\,\,\,L^\infty(\mathbb{R}^n)$$ and $$v_n\rightharpoonup v\,,\,\,\,\rm{in}\,\,\,L^2(\mathbb{R}^n)$$ What can I say of the convergence of the product $u_nv_n$? In particoular I want that $u_nv_n$ converges weakly in $L^2_{loc}(\mathbb{R}^n)$. I procced as follows: since $u_n$ converges strongly in $L^\infty$ it converges strongly in $L^2_{loc}$ and so we have a […]

Showing that $\cos(x)$ is a contraction mapping on $$

How do I show that $\cos(x)$ is a contraction mapping on $[0,\pi]$? I would normally use the mean value theorem and find $\max|-\sin(x)|$ on $(0,\pi)$ but I dont think this will work here. So I think I need to look at $|\cos(x)-\cos(y)|$ but I can’t see what to do to get this of the form […]