What is the 'implicit function theorem'?
Please give me an intuitive explanation of 'implicit function theorem'. I read some bits and pieces of information from some textbook, but they look too confusing, especially I do not understand why...
View ArticleParametrization and group operation
Let $A$ be a group of measurable transformation of $\mathcal{X} \subset \mathbb{R}^m$ into itself, such that each $a_\theta \in A$ is parameterized by $\theta \in \Theta \subset \mathbb{R}^d$.I have...
View ArticleDifference between $C_0(\mathbb{R})$ and $C(\mathbb{R})$
The definition for $C_0(\Omega)$ is continuous functions that vanish at the boundary of $\Omega$. When $\Omega = \mathbb{R}$ the boundary is an empty set thus we have $C(\mathbb{R}) = C_0(\mathbb{R})$....
View ArticleHow can I deal with floor functions inside a sum?
For context, imagine a sum along the variable $n$ whose terms $a_n = \lfloor nr\rfloor \cdot f(n)$ or $a_n = g(\lfloor nr\rfloor)$ for some value $r \in \mathbb{R}$. For example, I am working with the...
View ArticleQuestions about the Supremum and the Empty Set
In Rudin's Principles of Mathematical Analysis there are these two definitions:Definition 1.8:Suppose $S$ is an ordered set, $E \subset S$, and $E$ is bounded above. Suppose there exists an $\alpha \in...
View ArticleIf $\sup A < \inf B$ for sets $A$ and $B$, then there exists a $c \in R$...
I am reading Stephen Abbot’s Understanding Analysis and I am trying to complete the proof for Exercise 1.3.11 (b)If $\sup A < \inf B$ for sets $A$ and $B$, then there exists a $c \in R$ satisfying $...
View ArticleHow Holder continuity implies increasing homomorphism?
Let $X$ be a compact metric space. We define $\phi_t: X \rightarrow X$ such that $\phi_o = id, \phi_t \circ \phi_s = \phi_{s+t}$. Also, $\phi_t$ is a bijection with inverse $\phi_{-t}$. Also, each...
View ArticleEvery limit point is the limit of a sequence
Assume we have a metric space $X$, a subset $E\subseteq X$, and a limit point $p$ of $E$.Proofwiki and Rudin both "construct" a sequence that converges to $p$ using the fact that every neighborhood of...
View ArticleWhat are ways to compute polynomials that converge from above and below to a...
Main QuestionIn this question:A polynomial $P(x)$ is written in Bernstein form of degree $n$ if it is written as $P(x)=\sum_{k=0}^n a_k {n \choose k} x^k (1-x)^{n-k},$ where $a_0, ..., a_n$ are the...
View ArticleClosed form of an improper integral
Let $\alpha_1, \alpha_2 \in [0, \pi)$ be distinct real numbers (so that the functions $\sin(\theta + \alpha_1), \sin(\theta + \alpha_2)$ have no common zero) and let $0 < \gamma < 1$. Is there a...
View ArticleQuestion about behavior of integral of a function in $L^1$ with respect to...
Let $f:(0,\infty)\to \mathbb{R}$ function such that $f\geq0$,$$\int_{0}^{\infty}f(x)\mathrm{d}x=1\quad\text{and}\quad\lim_{x\to\infty}f(x)=0.$$Consider $0<t_1<t_2$. My question is: Is it possible...
View ArticleShow that, if $\sum (a_n)^2$ converges, then $\sum \frac{a_n}{n}$ also...
I figured out that $0 \leq \left( \sum \frac{a_n}{n} \right)^2 \leq \left( \sum a_n^2 \right) \left( \sum \frac{1}{n^2} \right)$. Then I concluded $\left( \sum \frac{a_n}{n} \right)^2$ converges, but...
View Article$f$ is integrable but $Hf$ is not
My real analysis homework has the following question:For $n\ge 3$ define $f=\sum_{n=3}^\infty \chi_{(n,n+{1\over n(\ln n)^2})}$. Show that $f$ is integrable but its associated Hardy Littlewood function...
View Article$\int_a^\infty x^r\sin(x)\,dx$ diverges for all $r>0$
How can I prove that for all $a,r>0$ the integral $\int_a^\infty x^r\sin(x)\,dx$ diverges?I first tried a limit comparison test with the divergent $\int_a^\infty x^r\,dx= \infty$.This leads to the...
View ArticleClarifying the concept of a sub-$\sigma$-algebra
From Definition 1 in Math 245A Note 1:Given two Boolean algebras $\mathcal{B}$, $\mathcal{B}'$ on ${X}$, we say that ${{\mathcal B}'}$ is finer than, a sub-algebra of, or a refinement of ${{\mathcal...
View ArticleConditions for existence of Fourier transform
What precisely is the set of necessary and sufficient conditions for the Fourier transform of a function $f : \mathbb{R} \mapsto \mathbb{C}$ to exist?Obviously there’s the sort of tautological answer...
View ArticleGriffiths and Harris Regularity Lemma II
I'm trying to make my way through Griffiths and Harris's proof of the Hodge Theorem in Principles of Algebraic Geometry. I've gotten through most of it just fine, but I've been stuck on the estimates...
View Articlethe validity of Bolzano's theorem
Sure this is a dumb question, but from what I know, Bolzano's theorem is a theorem so it should be true whenever the setting is the right one.however today in class we saw the case of $$...
View ArticleHow to prove Picard's existence and uniqueness theorem by Tonelli sequence...
$\qquad$First of all, this question for O.D.E. comes from an end-of-book exercise with no answer.$\qquad$Secondly, allow me to give the definitions of the relevant contents in the question to avoid...
View ArticleGiven $f : Q \to R$ s.t $f(x):= x \quad \forall x \in Q$ Can we find $g \in...
Given a function $f : Q \to R$where $Q$ is the set of all rational numbers and $R$ the set of real numbers.$$f(x):= x \quad \quad \forall x \in Q$$My question is can we find a continuous function $g :...
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