Articles of real analysis

Proof of strong maximum principle for harmonic functions

Let $u\in\mathscr C^2(U)\cap\mathscr C(\bar U)$ be harmonic in the non-empty open and connected set $U\subset\mathbb R^n$. If there exists a Point $x_0\in U$, so that $u$ has a local Maximum at $x_0$, then $u$ is constant in $U$. Proof: Let $x_0$ be an point in $U$ with $u(x_0)\geq u(x)$ for all $x$ in some neighbourhood […]

Non uniform continuity of a function and almost periodicity

We say that a continuous function $f:\mathbb{R}\to\mathbb{C}$ is almost periodic in the sense of Bohr if: For every sequence $(t’_n)_{n\geq0}$, there’s a sub-sequence $(t_n)_{n\geq0}$ such that $f(t+t_n)$ converges uniformly in $\mathbb{R}$ to a function $g(t)$ i.e. $$\sup_{t\in \mathbb{R}}|f(t+t_n)-g(t)|\to 0, \ \ when \ \ n\to +\infty.$$ This class of functions was proved to be the […]

Does $f_n(x) = \frac{x}{1 + nx^2}$ converge uniformly for $x \in \mathbb{R}$?

Does $f_n(x) = \frac{x}{1 + nx^2}$ converge uniformly to $f(x)$ for $x \in \mathbb{R}$? $\lim_{n \rightarrow \infty} f_n = f(x) = 0$ For $f_n(x) \rightarrow f(x)$ uniformly, I have to show that $sup_{x \in \mathbb{R}} |f_n(x) – f(x)| \rightarrow 0$ $sup_{x \in \mathbb{R}} |\frac{x}{1 + nx^2}| = \frac{1}{1 + n}$??? So I was thinking, $x […]

Show that a Lissajous curve has incommesurate frequencies iff it isdense in a rectangle

If we have a Lissajous curve given by the parametric equations $$x(t)=A\sin(\omega_x t)$$ $$y(t)=B\sin(\omega_y t + \delta),$$ how can we show that the curve is dense in the rectangle of sides $A,B$ if and only if $\omega_x$ and $\omega_y$ are incommensurate (i.e. their ratio is irrational)? What I tried: Suppose it is not dense, so […]

Integrability of a monotonic function and Darboux sum.

I have two questions on integrability. (a) Prove that $f(x)$ is integrable on $[a,b]$ if $f(x)$ is monotonic on the interval. (b) Define $f(x) = 1$ if $x$ rational, $0$ otherwise. Prove $f(x)$ is not integrable on $[0,1].$ For (a), there is a proof on my text book but it uses something called mesh which […]

Scale invariant definition of the Sobolev norm $\|\|_{m,\Omega}$ for $H^m(\Omega)$

I learned the following from Constantin and Foias’s Navier-Stokes Equations (Chapter 4): We say that a function of a bounded open set $\Omega\subset\mathbb{R}^n$, $c(\Omega)$, is scale invariant if $c(\Omega)=c(\Omega’)$ for all $\Omega’$ obtained from $\Omega$ by a rigid transformation and a dilation $x\mapsto\delta x$. Denote by $T_\delta$ the operation mapping functions defined on $\Omega$ to […]

A metric space is compact iff it is complete and totally bounded

I’m looking at this theorem from Marsden’s Elementary Classical Analysis, but there’s a part of the proof that I don’t understand. First I’ll state the proof of the direction that if it is complete and totally bounded then it is compact. The bold parts of the proof are what I don’t understand. Proof: Assume that […]

Prove that the length of a curve is given by integral

I have found the following problem in “Introduction to Analysis” by Rosenlicht. I am not sure if my solution is correct and I highlighted my uncertainties. First we have to show that the set $$S=\Bigg\{\sum_{i=1}^N d(f(x_{i-1}),f(x_i)):x_0,x_1,\ldots , x_N \mbox{ is a partition of } [a,b]\Bigg\}$$ is bounded from above and that the sequence of the […]

Show every subspace of $\mathbb R^n$ is closed with respect to the usual metric?

How do I see that every subspace of $\mathbb R^n$ is closed with respect to the usual metric $p(x,y) = x^Ty$ ? I’ve seen some sweet results regarding Hilbert Spaces $\mathcal H$, especially that for a subspace $S$, $(S^{\bot})^\bot = S$, and this is true (in the case $\mathcal H = \mathbb R^n$) if $S$ […]

Can absolute convergent series be expressed as sum of two series?

Let $C\subset \omega \bigwedge A\bigcap B = \emptyset \bigwedge A\bigcup B = C$. Let $\{x_i\}$ be a sequence of nonnegative reals. Suppose $C$ is infinite and $\sum_{i\in C} x_i$ converges. (Since it converges absolutely, it makes sense to define its sum in this way) Then $\sum_{i\in C} x_i = \sum_{i\in B} x_i + \sum_{i\in A} […]