Gradient of smooth function on unit sphere near the north pole
The two unit vectors$$\left\{\frac{(-y,x,0)}{\sqrt{x^2+y^2}}, \frac{(-xz,-yz,x^2+y^2)}{\sqrt{x^2+y^2}}\right\}$$form an orthonormal basis to the tangent space at $(x,y,z)\in \mathbb{S^2}\backslash...
View ArticleProve: $a_n \leqslant x$ $\forall_{n \geqslant 1} \Rightarrow \lim_{n \to...
I'm not sure about how to do the proof of this exercise of my math study. Its exercise 5.4.8 of Analysis I by Terence Tao:Let $(a_n) _{n=1}^{\infty}$ be a Cauchy sequence of rationals and $x \in...
View ArticleConfused about proof of theorem $ \lim_{x\to c} f(x) = L $ iff every such...
The theorem in my textbook says Assume $A \subseteq \mathbb{R}$, $f:A \rightarrow \mathbb{R}$, and $c$ is a limit point of A. Then $\lim_{x\to\ c} f(x) = L$ iff for every sequence $a_n$ from $A$ for...
View ArticleFunctions converge pointwise to a non-integrable function [closed]
Is there a sequence of continuous functions on $[a,b]$that converges pointwise to a continuous function $f$ which is not integrable?By Lebesgues theorem, the limiting function $f$ shouldn’t have...
View ArticleProving the Lp norm is a norm.
I want to prove the $L_p$ norm on continuous functions is in fact a norm. I have proven definitiveness and homogeneity but am struggling with the triangle inequality. I am using the fact that if the...
View ArticleSmooth function on bounded domain is Lipschitz
I am unsure, if my proof is correct. Would anybody please verify it?Let $f:\mathbb R^n \to \mathbb R$ be smooth, and let $U\subset R^d$ be a bounded domain, i.e., $U$ is open, connected and...
View ArticleLet $f \in C^\infty$, $f(0) = 0$ and $f(x) = \frac{1}{x}...
Let $f \in C^\infty$, $f(0) = 0$ and$f(x) = \frac{1}{x} \int_{\frac{x}{2}}^{\frac{3x}{2}} f(y) dy$$\overset{?}\implies$$f(x)=cx$I started by multiplying $x$ to the lhs. This gives $f(x)x =...
View Articlewhat do you call interchangeability $f(x,y)=f(y,x)$
Consider a real valued function $f(x,y)$.What is the formal mathematical notion of interchangeability (e.g., $f(x,y)=f(y,x)$)?
View ArticleQuestion about the $L^p$ being complete.
The inequality $(1+t)^p\leq 1+t^p$, which can be obtained by showing that $g(t)=(1+t)^p-t^p$ ($t>0$) is monotone decreasing implies that$$(a+b)^p\leq a^p+b^p,\quad 0<p<1,\,a,b\geq0.$$It then...
View ArticleIntuition for Conditional Expectation
It seems like NNT aka Nero in The Black Swan (2007) is giving the law of iterated expectations that involve filtrations in a heuristic way by matching the everyday usage of the word 'expect' with the...
View ArticleWhat are examples of everywhere defined unbounded linear operators on a...
Let $H$ be a Hilbert space.I know a lot of examples of linear operator $T$ on $H$ such that $D(T) \subsetneq H$ and $T$ is not bounded, and such kind of operators are very important in analysis of...
View ArticleWhat property causes the supremum to be adherent point for $\mathbb{R}$?...
In general, supremum is defined as the least upper bound. In the case of $\mathbb{R}$, we have an alternative characterization that $\text{sup} S $ is the supremum of a set $S$ if and only if for all...
View ArticleIs every set an open set?
I'm reading Rudin's "Principles of mathematical analysis", definition 2.18 stated that:"A neighborhood of $p$ is a set $N_{r}(p)$ consisting of all $q$ such that$d(p,q)<r$, for some $r>0$."This...
View ArticleSpectrum of the compact operator $Tx=(\alpha_n x_n)_n$ in $\ell^p$
Let $\alpha_n \in \mathbb C$ and $\lim_{n\to\infty}\alpha_n = 0$. Let $T$ be a linear continuous operator from $\ell^p \to \ell^p (1\le p\le \infty)$ defined by $$T((x_1, x_2, \ldots)) = (\alpha_1 x_1,...
View ArticleWhy is this function convex?
I am stuck in understanding the solution of this exercise. We have $f(x) = \begin{cases} 10 & x = 4 \\ x & 4 < x \leq 5 \end{cases}$ where also $f: [4, 5] \to \mathbb{R}$.The request asks to...
View ArticleLeast upper bound proof , $\{ x \in \mathbb{Q}: x^2 \le 2 \}$ [closed]
I am not super confident when it comes to proof so please bear with me. Is it possible to use proof by contradiction when trying to answer this question?Which of the following sets of real numbers have...
View ArticleNeed examples of problems of the form "Construct a function $f:[0,1]...
Let $I$ be the unit interval. I developed a method to simplify problems of the form "Construct a function $f:I \rightarrow I$ satisfying given properties". These properties should only be expressible...
View Articleproof of limits involving infinity using formal definition
I am reading my lecture notes on limits and stumbled across a 'solved' example problem.It concerns with the limit of $$ \lim_{x \to \infty} (x - \sqrt{x + 1}) = \infty$$Using the formal definition of $...
View ArticleNecessary condition for Gaussian KDE function to be nonnegative
Let $x_1, x_2, \dots x_n$ be fixed real numbers. Consider real numbers $v_1, v_2, \dots v_n$ such that $ v_1 + v_2 +\cdots + v_n > 0$. What condition do the points $v_1, v_2, \dots v_n$ need to...
View Article(Dis)prove $\frac{4(3\sqrt2-4)}{a+b+c+d}+\sum\limits_{cyc}\frac1{\sqrt...
An open problem from Art of Problem Solving (AoPS):If $a,b,c,d$ are positive real numbers such that$$\frac{1}{1+a}+\frac 1{1+b}+\frac 1{1+c}+\frac 1{1+d}=2,$$then prove or disprove$$\frac1{\sqrt...
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