Little-o allows two functions be little-o relatively to each other? Is not it...
I will use Wikipedia article about Big-O, little-o as a reference. Please note, I am asking about "little-o" exactly, not "big-o".Function $f(x)$ is $o(g(x))$at infinity, if there is real positive...
View ArticleConvergence or divergence of $a_{n+1}=a_n +\frac{a_{n-1}}{(n+1)^2}$
If $a_1=a_2=1$ and $a_{n+1}=a_n +\frac{a_{n-1}}{(n+1)^2}$ How to prove convergence of the sequence and its limit or divergence?It is easy to see that the sequence is always positive and by that one can...
View ArticlePrecise reference for Helmholtz decomposition
I am looking for a precise discussion of the Helmholtz decomposition. The usual statement I was able to find is something along the lineIf $\vec{v}$ is a smooth vector field vanishing at infinity, then...
View ArticleExistence of certain function satisfying certain conditions
I want to show that there does not exist $f ∈ C([0, 1], R)$ satisfying the following two conditions:(i) $\int_{0}^{1} f(x) dx = 1$; (ii) $\lim_{n\to\infty}\int_{0}^{1}f(x)^n dx = 0.$Suppose there exist...
View ArticlePassing from $\mathcal{D}$ to $\mathcal{S}$ using a density argument and...
I saw a proof for the following statement for the space of Schwartz functions $\mathcal{S}$:$$\varphi \in \mathcal{S}: \int \varphi = 0 \iff \exists \Phi\in \mathcal{S}: \Phi' = \varphi,$$which wasn't...
View ArticleA monotone sequence that diverges but has a convergent subsequence.
In Stephen Abbott textbook the solution states thatImpossible. This convergent subsequence would then be bounded; however, this would imply that the original sequence was also bounded. Because the...
View ArticleTotal variation and integral of $L^1$ function
Let $f \in L^1([a,b])$ with $[a,b]\subset \mathbb{R}$.Let$ F(x)= \int_{a}^{x} f(y) dy $, with $x\in [a,b]$.I’m really struggling to show that the total variation of $F$ coincide with...
View ArticleThe unit disc contains finitely many dyadic squares whose total area is...
Exercise 1.25.a in Pugh’s Real Mathematical Analysis states thatGiven $\epsilon > 0$, show that the unit disc contains finitely many dyadic squares whose total area exceeds $\pi - \epsilon$, and...
View ArticleInfinite Integration Issues
Let me preface this by saying that I know I'm doing something wrong, I'm just here to find out what exactly.We know that integrating a function yields constants. Something like $e^x$ yields $+c$ when...
View ArticleInverse operator of Laplacian
Suppose $u$ is a vector-valued function in $\mathbb R^n$, i.e. $u(x)=(u_1 (x), u_2 (x), \cdots,u_n (x))$ for $x \in \mathbb R^n$. How to derive: $$\partial_j \partial_k u = \partial_j \partial_k...
View ArticleShow that the $\sigma$-finiteness assumption of $\mu$ cannot be omitted in...
The Radon-Nikodym Theorem says the following:Theorem$\quad$Let $(X,\mathscr{A})$ be a measurable space, and let $\mu$ and $\nu$ be $\sigma$-finite positive measures on $(X,\mathscr{A})$. If $\nu$ is...
View ArticleShowing $\sum_{y \in Y} f(x_0, y) = \sum_{(x,y) \in \{x_0\} \times Y} f(x, y)$.
Let $X, Y$ be finite sets, and let $f: X \times Y \to \mathbb{R}$ be a function. I am trying to show that$$\sum_{y \in Y} f(x_0, y) = \sum_{(x,y) \in \{x_0\} \times Y} f(x, y) $$by using the following...
View ArticleComputing the inverse of Laplacian operator.
I am considering the following equation:$$f(t):=\int_\Omega[(I-t\Delta)^{-1}\Delta(I-t\Delta)^{-1}u]\cdot u\,dx$$where $u\in C_c^\infty(\Omega)$ and $t\geq 0$ a real number. $I$ is the identity...
View ArticleDefinition of the integral of a continuous function over a compact set when...
Let $h\ge 0$ be a continuous function defined on an open set $V\subset \mathbb{R^n}$.The integral of $h$ over a compact set $K \subset V$ can be defined as $$\int_K h(x)\,dx = \inf_{\substack{f \in...
View ArticleIs the image of a $G_\delta$ set under a continuous injection again a...
Question: Is the image of a $G_\delta$ set in $X$ under some continuous injection $f:X\to Y$ again a $G_\delta$ set in $Y$?I want to know this mainly for the case where $X=[0,1]$ and $Y=\Bbb R$. But, I...
View ArticleMollified tempered distribution and limit
Let $\mathscr{S}':=\mathscr{S}'(\mathbb{T}^d)$ be the space of tempered distribution on the $d$-dimensional torus $\mathbb{T}^d.$ Let $\rho,\phi \in C_c^\infty(\mathbb{R}^d,\mathbb{R})$ such that...
View ArticleConstruct a compact set of real numbers whose limit points form a countable set.
I searched and found out that the below is a compact set of real numbers whose limit points form a countable set.I know the set in real number is compact if and only if it is bounded and closed.It's...
View ArticleContinuous function which satisfies the Luzin N property, but which does not...
My question is: can we find the function $g\in{\rm C}([a,b])$ which satisfies the Luzin N property on $[a,b]$, but which does not satisfy the Banach S property on [a,b]? Here $[a,b]$ is a compact...
View ArticleVariant of Cramer's Large Deviation Theorem
Let $\Lambda(t) = \log \mathbb{E}[\exp(tX_1)]$ and define $\Lambda^*(x) := \sup_{t\in\mathbb{R}} (tx - \Lambda(t))$. Then according to Cramer's theorem,$$\lim_{n \to \infty} \frac{1}{n} \log \left( P...
View ArticleHow to construct a sequence $\{g_n\}$ of $\mathscr{A}$-measurable simple...
I am in the middle of proving a result. It would be unnecessary to type the whole thing out. I just want to ask one step where I got stuck on.So let $(X,\mathscr{A},\mu)$ be a measure space. Suppose...
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