$d\nu=d\lambda+fd\mu \implies d|\nu| = d|\lambda|+|f|d\mu$ Total variation...
On page 99 of Folland's Real Analysis, he states thatLet $\nu$ be a regular signed or complex Borel measure on $\mathbb{R}^{n}$, if $d\nu = d\lambda +fdm$ is a Lebesgue-Radon-Nikodym representation of...
View ArticleProduct of Sobolev functions $W_0^{1,p}$ and $W^{1,p}\cap L^\infty$
I have found two statements about products of Sobolev functions.In Evans & Gariepy's Measure Theory Section 4.2.2, $u,v\in W^{1,p}\cap L^\infty$ implies $uv\in W^{1,p}$.In Evans PDE Section 5.2.3,...
View ArticleAn interesting convex set property
Let $C$ be a nonempty convex subset of $\mathbb{R}^{2}$ and $f,g:[a,b]\rightarrow \mathbb{R}$ be two continuous functions such that $\left(f(t),g(t)\right)\in C$ for all $t \in [a,b]$.I want to prove...
View ArticleLower Semicontinuous Function = Supremum of Sequence of Continuous Functions
BackgroundI'm reading Cedric Villani's Optimal Transport: Old and New [1] and came across a result (below) I'm not quite sure how to prove. It is used to prove Lemma 4.3 and through my research, I've...
View ArticleWhy is the sequence of functions $f_n(x)=x^n$ not uniformly convergent in...
For the following sequence of functions and its limit function, we can see that $f_n(x)$ is clearly pointwise convergent$$f_n(x) = x^n\text{ }\forall x\in[0,1]\text{ and }\forall n\in\mathbb N^*\\f(x)...
View ArticleWhat are examples of functions with "very" discontinuous derivative? [duplicate]
Could someone give an example of a ‘very’ discontinuous derivative? I myself can only come up with examples where the derivative is discontinuous at only one point. I am assuming the function is...
View ArticleGeneralized Uniform Continuity?
I know that if $X$ is compact metric space then a continuous map $F:X\rightarrow Y$ is uniformly continuous and if $X$ not compact then it is not always true. But I was wonder if something for more...
View ArticleMean Value Theorem but with corner still holds?
As beautifully shown in https://en.wikipedia.org/wiki/Mean_value_theorem, if we have an everywhere differentiable function $f:\mathbb{R}\rightarrow\mathbb{R}$ and a secant between say $(a,f(a))$ and...
View ArticleDoes the Existence of Derivatives along All Smooth Paths Guarantee...
This question is inspired by other relevant questions on MSE regarding continuity of partial derivatives and differentiability, existence of partial derivatives and differentiability, and limits along...
View ArticleShow that $\ell^1$ is not complete with respect to the $\sup$ norm
I'm trying to make an example that shows $\ell^1$, that is the space of complex sequences that the sum of the norms of their components is finite, is not complete with respect to $\sup$ norm.And also a...
View ArticleIs there power series satisfying given the following four conditions?
Consider $g_n(x)=a_0+a_1x+\frac{a_2}{2!}x^2+\frac{a_3}{3!}x^3+\cdots$Suppose we would like to add constraints on its coefficients(1) $\sum_{k=1}^\infty\big|\frac{a_k}{k!}\big| \lesssim n^{-1/3}$(2)...
View ArticleInterior, boundary, derivate and closure of Intersection, union and product...
Assume the standard topology over $\mathbb{R}$ for this question.I also use:$\text{int}$ for the interior of a set;$\overline{A}$ for the closure of a set;$\partial$ for the boundary of a set;$A'$ for...
View ArticleDiscretisation of measurable function for simple functions approximation
In the classic proof of Rudin, Principles of mathematical analysis, Th 11.20, the discretisation of the range of the measurable function $f$ is with step $1/2^n$ for the interval [0,n], generating...
View ArticleFind $d_k$ that minimize $\sum_{k=1}^\infty...
Consider a strictly increasing sequence of strictly positive real numbers $d_k$. I would like to find their values so that they minimize the series$$\sum_{k=1}^\infty d_k\cdot...
View ArticleCutting across the graph of a function with a line through the origin
I have a question about a conclusion that is used in a proof/an example (Angus Taylor, General Theory of Functions and Integration, p. 304). We consider the set $L:=\{f\colon(0,1]\to\mathbb{R}:f(x)=ax,...
View ArticleDeducing existence and uniqueness of a solution from an inequality for the...
In the following, I am referring to this paper, p. 14, line (3.14):It is said that from$$\max\left(\sup_{n\geq 0}\Vert f^n\Vert_{2,\gamma}^2,\frac{2}{\varepsilon^2}\sum_{n=1}^\infty\Delta t\Vert...
View Article$\int_{0}^{\pi} \frac{dx}{(2 - \cos{x})^2}$ [duplicate]
For $a > 1$ we define the integral with the parameter$$F(a) = \int_{0}^{\pi} \frac{dx}{a - \cos{x}}$$I was able to find $F(a) = \frac{\pi}{\sqrt{a^2 - 1}}$. Any idea, how can I from here evaluate...
View ArticleProof that the angular measure is a measure.
Let $\mathbb{S}^1=\{x\in \mathbb{R}^2: \lvert x \rvert=1\}$. Consider $\mathcal{B}(\mathbb{S}^1)=\{B\cap\mathbb{S}^1:B \in \mathcal{B}(\mathbb{R}^2)\}$. I want to prove that there exists a measure on...
View ArticleKernel feature and derivative of kernel feature linearly independent?
Suppose we have a strictly positive definite symmetric kernel $k$ on an open set $\Omega\subset\mathbb R$. By "strictly" I mean that all kernel matrices $(k(x_i,x_j))_{i,j}$ with distinct $x_i$ are...
View ArticleShow that $\overline{K_r(x)} = \overline{K}_r(x)$.
Show that $\overline{K_r(x)} = \overline{K}_r(x)$.$K_r(x)$ denotes the open ball of radius $r$ centered at $x$$\overline{K_r(x)}$ is the closure of $K_r(x)$, which is the set of all points in $K_r(x)$...
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