Articles of analysis

A bounded sequence

I have a question please : Let $f:[0,2\pi]\times \mathbb{R} \rightarrow \mathbb{R}$ a differential function satisfying : $\displaystyle k^2\leq \liminf_{|x|\rightarrow \infty} \frac{f(t,x)}{x}\leq \limsup_{|x|\rightarrow \infty}\frac{f(t,x)}{x} \leq (k+1)^2$ Let $(x_n)\subset H^1([0,2\pi],\mathbb{R})=\lbrace x\in L^2([0,2\pi],\mathbb{R}),x’\in L^2([0,2\pi],\mathbb{R}),x(0)=x(2\pi)\rbrace$ sucht that $\|x_n\|\rightarrow \infty$ when $n \rightarrow \infty$ Why : the sequence $\left(\displaystyle\frac{f(t,x_n)-k^2 x_n}{\|x_n\|}\right)$ is bounded ? Is this answer given by :@TZakrevskiy true […]

Understanding a Proof for Why $\ell^2$ is Complete

Setting: Let $(x_n)$ be Cauchy in $\ell^2$ over $\mathbb{F} = \mathbb{C}$ or $\mathbb{R}$. I’m trying to show that $(x_n) \rightarrow x \in \ell^2$. That is, I’m trying to show that $\ell^2$ is complete in a particular way outlined below. I only used the first few steps of the proof because once I understand the third […]

Clarkson type inequality

Is it true that for $p\in (1,2)$ the following inequalities holds: $$ 2^{p-1} (|x|^p+|y|^p)\leq |x+y|^p+|x-y|^p \leq 2 (|x|^p+|y|^p)$$ for $x, y \in \mathbb{R}$ ? Thanks.

Is the subset $ ∩\mathbb{Q} ⊂ \mathbb{Q}$ closed, bounded, compact?

Letting $\mathbb{Q}$ be equipped with the Euclidean metric. What I can work out is that it is bounded as its contained in the closed ball of radius ${\sqrt2}/{2}$ centred at ${\sqrt2}/{2} $. Its not compact as it can be expressed as union of the two disjoint open sets $[0,{\sqrt2}/{2}) $and$ ({\sqrt2}/{2}, \sqrt2)$ (though I’m not […]

Computing fundamental forms of implicit surface

This question already has an answer here: About the second fundamental form 1 answer

Continuous functions with values in separable Banach space dense in $L^{2}$?

Let $[0,1] \subset \mathbb R$ be a the compact interval in the real numbers $\mathbb R$. We know that $C([0,1] \to \mathbb R)$ (the continuous function on $[0,1]$ with values in $\mathbb R$) are dense in $L^{2} ([0,1] \to \mathbb R)$ (the usual Lebesgue space). Now consider the Lebesgue space of functions on $[0,1]$, that […]

Analysing an optics model in discrete and continuous forms

A discrete one-dimensional model of optical imaging looks like this: $$I(r) = \sum_i e_i P(r – r_i)$$ Here, the $e_i$ are point light sources at locations $r_i$ in the object and $P$ is a point spread function that blurs each point. We can assume that $P$ is even, non-negative and has a finite extent, ie […]

A continuously differentiable function with vanishing determinant is non-injective?

(This question relates to my incomplete answer at Is the following true (for all n)? “If $f: \mathbb{R}^n \rightarrow \mathbb{R}^n$ is continuously differentiable and satisfies $\det(f'(x)) = 0$ for all $x$, then $f$ is not injective.” If so, what’s the most elementary proof you can think of? It is clearly true for $n=1$. In […]

Integral $\int \sqrt{x+\sqrt{x^2+2}}dx$

This question already has an answer here: Integrate $\int\sqrt{x+\sqrt{x^{2}+2}} dx$ . 3 answers

Open Cover for a Compact Subset

I am doing some extra exercises for an Analysis class, and I found this one. We haven’t seen much of what an open cover is, but I want to learn it. So, here it goes, and thank you everyone! Let compact subset $S \in \mathbb{R} $, with $\mathcal O$ as its open cover. Complete to […]