Theoretical foundations of calculus: limits, convergence of sequences, construction of the real numbers, least upper bound property, and related analysis topics such as continuity, differentiation, and integration through the fundamental theorem of calculus.

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“Simple” proof of Lebesgue's Differentiation Theorem for dimension 1?

Lebesgue Differentiation Theorem for $\mathbb{R}$: Let $f:[a,b]\to \mathbb{R}$ be intergable and $F(x)=\int_a^xf$. Then $F$ is differentiable almost everywhere in $[a,b]$ and $F'=f$ a.e. Is there a ...
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170 views

Functions from the Cantor set

Consider the Cantor set $\Delta\subset [0,1]$. Let $f\colon \Delta\to [0,\infty)$ be a continuous injection. Must $f$ be monotone on some uncountable closed subset of $\Delta$? Note that that van der ...
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152 views

A question about functions in $L^p(E)$

I've been working on a problem. I found a couple of related questions on the site, but I was having a little bit of trouble clarifying everything. The goal is to show that given $f\notin L^p(E)$, ...
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129 views

Proofing an inequality with a slowly varying function

I am working right now with "Independent and Stationary Sequences of Random Variables" from Ibragimov 1971. I am trying to understand the proof of the following Lemma (18.2.4): $h: \mathbb N ...
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640 views

Distinct eigenvalues for integral operator?

Is there some sufficient condition on the kernel $K$ of a (say, finite rank, to simplify) integral operator $$ \mathcal K:f(x)\in L^2(\mathbb R)\mapsto \int K(x,y)f(y)dy $$ so that it has all its ...
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605 views

one to one mapping between the floor function and the Riemann prime counting function

We have the following 'transform' of a real valued, piecewise continuous function $f(x)$ : $$T[f(x)]=1+\sum_{n=1}^{\infty}\int_{\mathbb{R}^{n}_{+}}f\left(\frac{x}{\Lambda _{n}} \right ...
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270 views

Fourier dimension of sets of positive Lebesgue measure

Let $K$ be a compact set in $\mathbb{R}$ with positive Lebesgue measure. My question is whether there exists a probability measure $\mu$ supported on $K$ such that $\hat{\mu}(\xi)$, the ...
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409 views

German Analysis Texts

My question is somewhat related to this one but is somewhat more specific. Since a lot of good mathematics is written in German, I have decided to start developing my German reading abilities. So far, ...
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245 views

Points in the plane at integer distances

Does there exist a set of $n$ points $p_1,p_2,...,p_n$ in the plane, all at mutual integer distances to each other, and an $e>0$, such that the following statement holds: For all $a,b$ with ...
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287 views

What are the conditions sufficient and necessary on $g(t)$ for the Dirichlet integral to be equal to $\frac{\pi}{2} g(0+)$?

Dirichlet Integral of a function $g\colon \mathbb{R} \to \mathbb{R}$ is defined as $$ DI(\alpha) = \int_0^{\delta} g(t) \frac{\sin(\alpha t)}{t} dt$$ assume $\alpha \in \mathbb{N}$ For the equality ...
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143 views

Convergence check (No steps/solutions/proofs please)

I just wish to check that I have got these right. Please just indicate whether the ans are right -- please don't show steps (I wish to figure those out myself). Given $\phi_n (x)=n^k(1-x)x^n$ where ...
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187 views

Intuitive test of convergence

Are there any intuitive tests that might help one decide whether a sequence of functions converges / converges uniformly? For example, an intuitive test I have recently realized for uniform continuity ...
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208 views

How should I understand $u_{\infty}$ in this theorem?

I learned the extension of Green's formula to unbounded domains in Kress's Linear Integral Equations (p.71): Theorem 6.10 Assume that $D$ is a bounded domain of class $C^1$ with a connected boundary ...
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192 views

A relation between permanents and determinants

I have skimmed this video that I found on mathoverflow: http://tube.sfu-kras.ru/video/407?playlist=397 At about 15:05 the lecturer wrote down an equality $\sum F(m_1, \ldots, m_m)z^{m_1}\ldots ...
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40 views

Can we use sequences to test continuity of a weak$^*$-continuous operator?

Let $X,Y$ be Banach spaces. Now assume we have a map $T:X'\rightarrow Y'$ where $X'$ and $Y'$ are equipped with the weak$^*$ topology and not the norm topology. Can I infer from this that an operator ...
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61 views

minimize the area of convex hull of sum of 3 balls

How should we place 3 balls $B_1,B_2,B_3$ on the plane, if we want to minimize the area of convex hull of $B_1\cup B_2\cup B_3$ ? Balls can have boundary common points only -- the intersection of any ...
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54 views

Bartoszyński's results on measure and category and their importance

I have seen this interesting paragraph on a talk page of the Wikipedia article about Polish mathematician Tomek Bartoszyński: Tomek's paper "Additivity of measure implies additivity of category, ...
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58 views

Theorem 3.7 in Baby Rudin: The subsequential limits of a sequence in a metric space form a closed set

Here's Theorem 3.7 in the book Principles of Mathematical Analysis by Walter Rudin, 3rd edition. The subsequential limits of a sequence $(p_n)$ in a metric space $X$ form a closed subset of $X$. ...
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53 views

Rational Function

The problem is, Find all continuous functions $f: \mathbb{R} \to \mathbb{R}$ such that if $x-y$ is rational, $f(x)-f(y)$ is rational. I was thinking that only constant functions would work. How ...
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33 views

Are physical/material/dimensional/temporal explanations of Banach-Tarski necessarily irrelevant?

I recently re-reviewed some of my undergrad analysis text and read the sketch of the proof of Banach-Tarski presented on Wikipedia, starting with a proof that the free group with two generating ...
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84 views

Using the IVP definition of $\cos$ and $\sin$, how can we show that $\cos^2(x)+\sin^2(x) = 1$ without any “magic”?

One way to define the exponential, hyperbolic and circular functions is to assert that they're the unique solutions to certain IVP systems: The exponential function: $$\exp'(x) = \exp(x), \qquad ...
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37 views

$L^{2}$ convergence of sequence $|u_{j}|^{p}\nabla u_{j}$

Suppose a sequence $\{u_{j}\}$ is bounded in the Sobolev space $H^{2+\epsilon}(\Omega)$, for $\epsilon>0$, where $\Omega$ is say a bounded, $C^{\infty}$ domain. Here, the fractional space ...
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55 views

The space of test-functions carries any other structure on it?

I'm starting to study distributions and on the lecture notes I'm reading the author defines a test-function as a function $f : U\subset \mathbb{R}^n\to \mathbb{R}$ which is infinitely differentiable ...
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83 views

Inequalities involving polynomials with combinatorial coefficients

For all non-negative integers $i$ and $j$ such that $j\leq i$, define the array of polynomials $$p_{ij}(z):=\sum_{h=(j-1)_+}^{i-1} {i\choose h}{i-j\choose{i-h-1}}z^h,$$ where $(a)_+=\max\{a,0\}$ (we ...
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31 views

Tensor product of $L^1([0,1],\omega)$ with some Banach space

We know that for any Banach space $E$ if $\hat{\otimes}$ denotes projective tensor product then $L^1([0,1])\hat{\otimes} E$ is isomorphism isometric with $$L^1([0,1],E)=\{f:[0,1]\to E:\ \ ...
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50 views

Second derivative test for $f(0)=f'(0)=0$ and $f''(0)=2$

Let $f\in C(\mathbb{R}^1)$ with $f(0)=f'(0)=0$ and $f''(0)=2$. Prove that $0$ is strict local minima of $f$. Proof: Since $f''(0)=2$ then $\exists\delta>0$ such that for any $t\in \mathbb{R}^1$ ...
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208 views

How did he use Gronwall Lemma??

I´ve got these lines from an article: ( where $b:\mathbb{R}_+\to \mathbb{R}_+$ is non-decreasing and $(X_t)$ is an $\mathbb{R}_+$-valued process - it doesn't matter very much, I guess, anyway-.) ...
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72 views

Is the sum of infinitely many open sets open?

Let $X$ be a locally convex space (or, in particular, a normed space). Let $(O_n)_{n=1}^\infty$ be an infinite sequence of non-empty open sets in $X$ such that the sum $\displaystyle\sum_{n=1}^\infty ...
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61 views

Lebesgue versus Riemann integrable

Can a Lebesgue measurable function be modified on a set of first category so as become continuous except on a set of Lebesgue measure zero? OR Can a Baire-measurable function be modified on a set of ...
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50 views

Looking for a function that satisfies some kind of mean value property

Given $a<b\in (0,1)$ and $\delta<1/2$, I need to find an integrable function $\gamma :(a-\delta,b+\delta)\to [0,1]$ such that $$\frac{1}{2\delta}\int_{x-\delta}^{x+\delta}\gamma(y)\; ...
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85 views

Example of subsets in plane with continuous bijective mapping between them

The question is from C. Pugh's Real Analysis: Construct nonhomeomorphic connected, closed subsets A, B $\subset$ $\mathbb{R}^2$ for which there exists continuous bijections $\;f: A \to B$ and $\;g: ...
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31 views

Function $f:\mathbb{R}^k \rightarrow \mathbb{H}$ is differentiable if partial derivatives are continuous.

Let $f: \mathbb{R}^k \rightarrow \mathbb{H}$ be a function from the euclidean space to a Hilbert space. Is it true that if the partial derivatives are continuous, then $f$ is differentiable? (the ...
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34 views

Help to understand a Lemma about 'supremum of a family of measures'

I read the following Lemma from a paper but I can't understand the proof. Please help! Lemma: Let $\mu$ be a positive measure defined on the family of open subsets of $\Omega$, which is ...
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26 views

Literature on the convergence of $x_{n+1} = f(x_n)$ in general

When faced with a recurrence of the form $x_{n+1} = f(x_n)$, my toolbelt for proving convergence is very limited: if $f$ isn't $k$-lipschitzian with $k<1$, and/or if I can't find some complete set ...
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64 views

If $m(A)=0$ and $f:[0,1]\rightarrow [0,1]$ has bounded derivative then $m(f(A))=0$

Let $A\subset [0,1]$. Show that if $m(A)=0$ and $f:[0,1]\rightarrow [0,1]$ has bounded derivative then $m(f(A))=0$. ($m$ denotes Lebesgue measure on $[0,1]$) If $m(A)=0$ then for any ...
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46 views

is there something wrong with the Fourier transform coefficients?

define the Fourier transform and its inverse $$\hat{u}(y)=\frac{1}{(2\pi)^{n/2}}\int_{\mathbb R^n}e^{-ixy}u(x)\,dx$$ $$\check{u}(y)=\frac{1}{(2\pi)^{n/2}}\int_{\mathbb R^n}e^{ixy}u(x)\,dx$$ then ...
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44 views

Direct sum of $\mathbb{R}(B)$-modules consisting of all cross sections.

Let $B$ be a Tychonoff space and let $\mathbb{R}(B)$ denote the ring of continuous real valued functions on $B$. For any vector bundle $\xi$ over $B$ let $S(\xi)$ denote the $\mathbb{R}(B)$-module ...
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61 views

$\| f \|_{BV} = | f(a) | + V_{a}^{b} f$ defines a norm in the space $BV[a,b]$

I'm learning about functions of bounded variations and need to verify my work for this problem: Show that $\| f \|_{BV} = | f(a) | + V_{a}^{b} f$ defines a norm in the space $BV[a,b]$. My ...
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46 views

Computing the limit of this integral,

This is Part 6 (last part) of a problem statement of an old comprehensive exam question that I am working on. It asks to evaluate $$\lim_{r_0 \to 0} \int_{-\infty}^{\infty} ...
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85 views

Minimal conditions to show $\sum \rho_{ij} \Psi_{ij} s_i s_j < \sum s_i s_j $

Consider a sequence of real number $\{s_i\}_{i\leq n}$. Now consider the real numbers $F$, $G$ and $\alpha$ defined below $$F= \sqrt{ \left( \sum ~\rho_{ij} ~\Psi_{ij}~ s_i ~s_j \right)^+}, $$ $$G = ...
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68 views

The proof of Hahn Decomposition Theorem in Folland (real analysis)

On p.86, the following is Folland's proof of Hahn: Before the red line, it is not hard to understand and I think the proof is enough for showing $N$ is negative. Since it proves that: $N$ ...
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65 views

Rigorous proof of this limit

I have shown that the function $$f(x):=\int_{[-\pi,\pi]^n} \frac{e^{-i\langle k,x \rangle}}{1-\frac{1}{n} \sum_{i=1}^n \cos(k_i)}dk$$ exists everywhere for $n \ge 3$. Now, I want to show that ...
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61 views

Show that if $\int_I f=0$ for all interval then $f=0$

Let $f$ integrable on $\mathbb R$ and continuous. I have to show that if for all interval $I\subset \mathbb R$, $$\int_If=0$$ then $f=0$. My attempts Suppose $f\neq 0$. Then, there is a $c\in\mathbb ...
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152 views

Find the complex (or real) roots of $e^{\frac{3 x}{2}}+2 \cos \left(\frac{\sqrt{3} x}{2}\right)$

Define for natural $n\geq 2$ $$G(x,n)= \sum _{k=0}^\infty \frac{x^{k n}}{(k n)!}= \frac{\sum _{k=0}^{n-1} e^{x e^{\frac{2 i \pi k}{n}}}}{n}= G(x e^{\frac{2 i \pi}{n}},n)= \prod_{m=1}^\infty ...
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55 views

Why do these Integration-by-Parts Evaluation Terms Vanish?

The Associated Legendre operator is $$ L_mf = -\frac{d}{dx}\left((1-x^{2})\frac{df}{dx}\right)+\frac{m^{2}}{1-x^{2}}f, $$ where $m$ is a positive integer. For the purposes here, define ...
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65 views

Codimensions of $\mathbb{Q}$-subspaces of $\mathbb{R}$

Under the Axiom of Choice, we can pick a Hamel basis $H$ for $\mathbb{R}$ as a vector space over $\mathbb{Q}$. Adjoining all but some elements of $H$ to $\mathbb{Q}$ shows that for any cardinality ...
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170 views

trace norm and tensor product

Let $(M_n (\mathbb{C}), n\|.\|)$ , $(M_n (\mathbb{C}), n\|.\|)$ and $(M_{nm} (\mathbb{C}), nm\|.\|)$ be three Banach algebras. where $$\|A\| = \mathrm{tr}\sqrt{(A^* A)}. $$ What is the norm of $\phi$ ...
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107 views

Reference request: are levelsets/levelspaces (as defined here) used anywhere in mathematics?

Thinking about unimodal distributions in multiple dimensions, I came up with the following. Given a function $f : Y \leftarrow X$ where $X$ is a topological space, define that a levelset of $f$ is a ...
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46 views

Show that $(3x_{n}+4y_{n})$ is also Cauchy sequence.

Show that if $(x_{n})$ and $(y_{n})$ are Cauchy sequences in $X$, then the sequence $(3x_{n}+4y_{n})$ is also Cauchy sequence using the definition of a Cauchy sequence. Attempt Let $\epsilon > 0$ ...
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75 views

Is there any flaw in my proof of Darboux's Theorem?

Darboux's Theorem. Let $I$ be an open interval, and let $f : I \to R$ be a differentiable function. If $a$ and $b$ are points of $I$ with $a < b$ and if $y$ lies between $f' (a)$ and $f' (b)$, ...