Articles of real analysis

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} […]

A strongly pseudomonotone map that is not strongly monotone

Let $K$ be a closed convex subset in $\mathbb{R}^n$ and $F: K\rightarrow \mathbb{R}^n$. We say that $F$ is strongly monotone on $K$ if there exists $\gamma>0$ such that $$ \left<F(y)-F(x), y-x\right>\geq \gamma\|y-x\|^2, \quad \forall x,y\in K. $$ $F$ is strongly pseudomonotone on $K$ if there exists $\gamma>0$ such that $$ \left<F(x), y-x\right>\geq 0 \Longrightarrow \left<F(y), […]

$\lim_{x\to\infty}\int_x^{x+1}f(y)dy=0$ implies $\lim_{x\to\infty}\frac{\int_0^{x}f(y)dy}{x}=0$

Let $f$ be a non negative continuous function on $[0,\infty)$ such that $$\lim_{x\to\infty}\int_x^{x+1}f(y)dy=0.$$ How do we prove that $$\lim_{x\to\infty}\frac{\int_0^{x}f(y)dy}{x}=0.$$ If we see this question using the primitive function, do we have the following result for a continuous function $F$: $$\lim_{x\to\infty}F(x+1)-F(x)=0$$ implies that $$\lim_{x\to\infty}\frac{F(x)}{x}=0.$$

$f: \mathbb{R}^2\to \mathbb{R}^2$ is differentiable, and satisfies an inequality that involves its partials – show that f is a bijection.

Suppose that $f: \mathbb{R}^2\to \mathbb{R}^2$ is differentiable, and the partial derivatives of the components $f_1$, $f_2$ satisfy $$max(|\frac{\ df_1}{dx} -1|, |\frac{df_1}{d_y}|, |\frac{df_2}{d_x}|, |\frac{df_2}{d_y}-1|) <10^{-10}.$$ Prove that f is a bijection. Note: f is not assumed to be continuously differentiable. Any ideas on how to tackle this problem? We don’t have an explicit function given for […]