"Continuity" in a metric space
I just learned about metric spaces and their basic properties and definitions, such as the definition of a continuous function that maps two metric spaces. However, when I say "continuity" on a metric...
View ArticleThe set of all rational points in the plane is a countable set
From Kolmogorov's Introductory Real Analysis. I am doing some self-study and would like some feedback on whether my proof is correct.I am using that the set of rational numbers is countable as given,...
View ArticleRiemann sum without the boundary sets
Let $X\in \mathbb R^m$ be J-measurable set and $f:X\to \mathbb R$ an integrable function. I would like to prove this Riemann sum formula still holds$$\int _Xf(x)dx=\lim_{|D|\to 0}\Sigma(f;D^*)$$even we...
View ArticleDoes the expectation of eigenvectors of resolvents of random matrices vary...
Suppose X is a real $p\times n$ random matrix and $Y=XX^T/n $. The resolvent of Y is$$R(z)=(Y-z)^{-1},$$ where $z$ is a complex number. Since the Stieltjes transform of a random matrix can be written...
View ArticleProve that the adjoint of $Au=\sum_k \alpha_k\langle u,e_k\rangle e_k$ is...
Let $H$ be a separable complex Hilbert space with orthonormal basis $\{e_k; k \in\mathbb{N}\}$. Let $(\alpha_k)$ be a given sequence of complex numbers and let $A$ be the associated multiplication...
View Article$\lim F(a) = +\infty$ where $F(a):=\int_B \frac{1}{|z-a|}...
Let $B=\{(x,y)\in\mathbb{R}^2: x^2+y^2<1\}$. Let $z$ denote $(x,y)$, and let $$F(a):=\int_B \dfrac{1}{|z-a|} (1-|z|)^{-1/2}d xdy,\quad\forall a\in B.$$I want to verify that $\lim_{a\rightarrow...
View Article$X$ compact, limit of $f$ exists for all $p\in X$ then $f$ uniformly...
For simplicity, let's take our compact set $X$ to be a subset of $\mathbb{R}$. Define $f:X\to\mathbb{R}$ to have a limit at every point in $X$. Then $f$ is uniformly continuous on a set $X$ minus a...
View Article(Possible) Solution to Putnam 1976 A6 $f : R \mapsto [-1,1]$ twice...
I was hoping someone could take the time to correct my proposed solution. Thanks.Putnam 1976 A6:Let $f : R \mapsto [-1,1]$ be twice differentiable and satisfy $(f(0))^2 + (f'(0))^2 = 4$. Show that...
View ArticleFinite dimensional $l^p$ Norm is logarithmically convex.
I am working trough this book, where in lemma 1.6 (proof left to the reader), the asked to show that the map $p \mapsto \log \|X\|_{p,J}^p$ is convex. Here $\|X\|_{p,J}$ is the $p$-variation of a...
View ArticleUniform convergence and differentiation; difference between two theorems?
I am comparing two theorems in two different texts on uniform convergence and differentiability; the first one is from these lecture notes (page 9 in the pdf), the second from Rudin's PMA.Here's the...
View ArticleA more general version of Folland, "Real Analysis", Theorem 1.14 [duplicate]
Let $ X $ be a set. Let $ \mathcal E $ be a semi-ring of ("elementary") subsets of $ X $, i.e. assume that$ \emptyset\in \mathcal E $;$ \mathcal E $ is closed for finite intersections;the set...
View ArticleShow a function defined by an integral is differentiable [closed]
Let $f=f(x,y)$ be given by: $$f(x,y)=\int_x^{x+y}\int_{x-y}^ye^{s^2+t^2}dsdt$$Show that $f$ is differentiable and calculate $\nabla f(x,y)$Usually I would write my attempt, however, I have no idea what...
View ArticleDoes $\sum_{k\geq0}\frac{z^k}{k!^s}$ only have negative real roots?
It might be complicated to decide whether or not a given entire function has only real zeros. So, I'll focus on special examples. It is known that the following function given by the infinite...
View ArticleProve that the limits of these two recurring sequences is pi.
I'm trying to prove that these two sequences $a_n$ and $b_n$ converge to $\pi$, but cannot find a method of doing so. The two sequences are defined as such:$a_0=2\sqrt3, b_0=3$$a_n=...
View ArticleWhich proof is more general? (Derivative of Series is Series of Derivatives)
Let $\Omega \subseteq \mathbb{R}$ be open and $(f_n)_n$ a sequence in $C_b^1(\Omega, \mathbb{C})$ (the space of $C^1$-functions $g$ such that $g$ and $g'$ are bounded). Equipping this space with the...
View ArticleUniqueness of Caratheodory Extension
I am trying to prove the following:Given a pre-measure $p : R \rightarrow [0,+\infty]$ where $R \subset 2^X$ is a semi-ring, define the outer measure $u^{\ast}(E) = \inf\{\sum_{j=1}^{\infty} p(A_j) : E...
View ArticleConstructing a pointwise convergent sequence of continuous functions
Given two sequences of real numbers $(\epsilon_n)\subseteq (0, 1)$ and $(s_n)_{n\in \mathbb{N}} \subseteq (1, \infty)$ such that $\lim_n r_n=0$ and $(s_n)_{n\in \mathbb{N}}$ is unbounded, is it...
View ArticleIf $f:\mathbb{R}\to\mathbb{R}$ is continuous, does there exists a nonzero...
Is the following proposition true or false?If $f:\mathbb{R}\to\mathbb{R}$ is continuous, then there exists a not-everywhere-zero (i.e. $g\not\equiv 0$)continuous function $g:\mathbb{R}\to\mathbb{R},\ $...
View ArticleProof of Ostrogradsky's Formula-- Integration of rational functions
Ostrogradsky method is in general formula $$\int \frac{P(x)}{Q(x)}=\frac{P_1(x)}{Q_1(x)}+\int \frac{p(x)}{q(x)}dx$$Where $\frac{P(x)}{Q(x)}$ is a simplest rational function; $q(x)$ is a polynomial with...
View ArticleIs density preserved under continuous functions? [duplicate]
This is a generalization of a previous question, here: Are the set of squares (and also higher powers) of positive rationals dense in the positive reals?. Suppose $S$ is a dense subset of the reals,...
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