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Question about the proof of "If K is a compact set of the metric space Ω,...
I am reading the proof of the Theorem: If $K$ is a compact set of the metric space $\Omega$, then $K$ is closed, and I encounter a problem.Here is the proof in the book What I don't understand is the...
View ArticleHow to show that the inf / sup of the set of the $n$-th powers of the...
Let $A$ be a nonempty, bounded (from above) set consisting of non-negative real numbers only, let $n$ be a given positive integer, and let sets $B$ and $C$ be defined as follows:$$B := \left\{ a^n...
View ArticleInvestigate whether $f_n (x)=\dfrac{nx}{1+n^3x^3}$ on $[0,\infty),n\in...
Investigate whether $f_n (x)=\dfrac{nx}{1+n^3x^3}$ on $[0,\infty),n\in \mathbb N$ converges uniformly or not.Obviously, $f_n(x)$ pointwise converges to $0$. Since $f_n(0)=0$ when $x=0$ and...
View ArticleA Hölder norm of square root of a $C^2$ function
BackgroundI am reading a proof of the Calabi-Yau theorem from these notes. In page 15 he claims the following statement without proof (calling it elementary): Let $M$ be a compact manifold. There...
View ArticleMinkowski inequality of infinite sum
For $1\leq p <\infty,$Given $\{f_n\}^{\infty}_{n=1}$ be a sequence of function in $L^{p}(\mathbb{R}).$Show that $\left\Vert \sum\limits_{n=1}^\infty f_n\right\Vert_p \leq \sum\limits_{n=1}^\infty...
View ArticleProving the set of all cluster points is closed
if $A$ is a subset of $\mathbb{R}^n$. Can I prove that "the set of all cluster points of A is closed" using the following:A closed subset is a subset that contains all its limit points. Every cluster...
View ArticleQuestion about proof of Lebesgue Decomposition Theorem for the case of...
I am asked to proof the following Lebesgue Decomposition Theorem for the case of $\sigma$-finite positive measure:Lebesgue Decomposition Theorem$\quad$ Let $(X,\mathscr{A})$ be a measurable space, let...
View ArticleApproximation of the integral of a 1-periodic $L^2$-function
Let $\alpha $ be a fixed irrational number. For a function $g:\Bbb R\to\Bbb C$, define $$g^*(x)=\sup_{N\geq 1} \frac{1}{N} \sum_{n=1}^N |g(x+\alpha n)| ,$$and assume that there is a constant $C>0$...
View ArticleDefinition of continuity rephrased epsilon delta (need help checking my...
I am trying to rephrase the definition of continuity to see if I have the right intuition. Please comment on how I might be wrong or misunderstanding something:A function f is continuous at a point...
View ArticleProving Closure Under Multiplication in a Set with Specific Properties
I encountered a problem where I needed to prove that a set ( A ) is closed under multiplication, given the following properties:$0 \in A$$1 \in A$If $x, y \in A$, then $x - y \in A$If $x \neq 0$, then...
View ArticleCan someone prove $\int_a^bf(x)dx=\frac{1}{n}\int_{na}^{nb}...
Question:Hi everyone,I was working on a problem and came across the idea that the integral of a function $f(x)$ from $a$ to $ b $ can be equal to the integral of the function $...
View ArticleQuestions about a proof of Peano's existence theorem
My lecturer presented a proof of a version of Peano's existence theorem using the Euler method. I have trouble understanding some particular steps. The statement is the following:Let $U \subseteq...
View ArticleThe infinite product of the expression.
I have come across the following expression, but I do not know how to prove it. I have tried using mathematical induction, but I haven't achieved anything, and it seems very complicated.$$e^{-\frac{\pi...
View ArticleSymmetric differentiation of real functions
Let a function $f:\mathbb{R}\longrightarrow\mathbb{R}$ and $x\in \mathbb{R}$.$f$ is symmetrically continuous in $x$ if $\lim\limits_{h\to 0} [f(x+h)-f(x-h)] = 0$.$f$ is symmetrically differentiable in...
View ArticleCould we approximate $\int_0^1\frac{1}{x^4}dx$ using a Riemann sum?
We know that in 1 dimension, the integral $\int_0^1\frac{1}{x^4}dx$ is not finite. But could we approximate this integral using a Riemann sum?In particular, if we divide the interval $[0,1]$ into...
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Rudin $8.18$ theorem.
There are necessary definitions and theorem for the proof:here is the theorem:If $1\lt p \lt \infty$ and $f \in L^{p}(R^k)$ then $Mf \in L^{p}(R^k)$there is its proof:Since $Mf = M(|f|)$ we may assume,...
View ArticleProve that $\int\limits_0^1 \bigg | \frac{f''(x)}{f(x)} \bigg| dx \ge...
Let $f:[0,1]\to \mathbb{R}$ be a $C^2$ class function such that $f(0)=f(1)=1$ and $f(x)>1,\forall x\in (0,1)$.Prove that $$\int\limits_0^1 \bigg | \frac{f''(x)}{f(x)} \bigg| dx \ge...
View ArticleUse a Riemann sum to approximate the integral...
Consider the function $f:[-1,1]\setminus\{0\}\to \mathbb{R}$ given by $f(x)=\frac{1}{|x|^{2.5}}.$ For dimension $d=1,$ Consider the integral below:$$\int_{-1}^1\frac{1}{|x|^{2.5}}(e^{-i\pi\omega\cdot...
View ArticleCan reparameterization make Cramer-Rao bounds tight?
Given a family of distributions parametrized by $\theta$ for which Cramer-Rao bounds on variance for a (biased or unbiased) estimator of $\theta$ exist, these bounds may be unattainable. In the proof...
View ArticleProve the inverse Fourier transform of Gaussian kernel is Gaussian...
How do I prove$$p(w) = \int_{\mathbb{R}^D} e^{-jw^T \delta} k(\delta) d\delta $$(i.e. $p(w)$ is the inverse Fourier transform of $k(\delta)$) where $p(w)$ is the multivariate Gaussian...
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