Articles of real analysis

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 […]

Determining whether $\int_{0}^{\infty} \frac{x \sin(x)}{1+x^2}dx$ converges and converges absolutely

I would like to check whether $$\int_{0}^{\infty} \frac{x \sin(x)}{1+x²}dx$$ converges and converges absolutely. I have a feeling that neither is true, however none of the methods known to me seem to help. I struggle to find a lower estimate for the function. Any hints and help welcome. I tried using $$\frac{x \sin(x)}{1+x²}\leq \frac{x \sin(x)}{x²}=\frac{ \sin(x)}{x}$$ […]

Behavior at $0$ of a function that is absolutely continuous on $$

The function $f$ on $[0,1]$ is absolutely continuous on $[\epsilon,1]$ for $0<\epsilon<1.$ I further have that $$\int_0^1x|f'(x)|^pdx<\infty.$$ I’m trying to show that $$ \lim_{x\to 0}f(x)\ \text{exists and is finite}\qquad \text{if}\ p>2, $$ $$ \frac{f(x)}{|\log x|^{1/2}}\to 0\ \text{as}\ x\to 0\qquad \text{if}\ p=2,\ \text{and} $$ $$ \frac{f(x)}{x^{1-\frac{2}{p}}}\to 0\ \text{as}\ x\to 0\qquad \text{if}\ p<2. $$ (Of course, for […]