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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My (simpler) proof of the divergence of a harmonic series.

Let $H=1+\frac{1}{2}+\frac{1}{3}+\cdots$ . Proving that it diverges this is what I did. I supposed that the series converges to $H$, : $$H\geq1+\frac{1}{2}+\frac{1}{4}+\frac{1}{4}+\frac{1}{6}+\frac{...
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154 views

Can one use $e^n$ instead of $2^n$ in Cauchy condensation test?

Cauchy condensation test is useful for testing the convergence of infinite series. The test is stated here as follows: for a positive non-increasing sequence $f(n)$, the sum $\sum_{n=1}^\infty f(n)$ ...
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94 views

What kinds of structures support integration?

I am doing topology which neatly generalizes analysis, which led me to wonder naturally about generalizations of calculus. Specifically I'm interested in knowing what is required of a mathematical ...
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232 views

lim sup intuition for a sequence of sets

lim sup of a sequence of sets $(E_n)$ is defined as $$\bigcap_{n = 1}^{\infty}\bigcup_{k = n}^{\infty} E_k$$ and this means that an element ins lim sup $E_n$ is a member of infinitely many of the ...
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279 views

Dyadic approach to Marcinkiewicz interpolation for Lorentz Spaces

In Exercise 21 , in a note for professor Terrence Tao on his own blog http://terrytao.wordpress.com/2009/03/30/245c-notes-1-interpolation-of-lp-spaces/ Exercise 21 Suppose we are in the ...
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249 views

Is there an orthonormal basis for $L_2[0,1]$ consisting of convex functions?

Is there an orthonormal basis $\{\phi_{\alpha}\}$ for the space $L_2[0,1]$ of square-integrable functions from $[0,1]$ to $\mathbb{R}$ such that every $\phi_{\alpha}$ is convex? Edit: A helpful ...
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166 views

Existence of smooth extension of a function defined on a closed interval

Suppose $ f: [0,1] \rightarrow \mathbb{R}$ is a function such that derivatives of all orders exist ( at the end points of the interval the appropriate one-sided derivatives exist) and are continuous $ ...
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296 views

Properties of the projection onto a nonconvex set

Consider a set $\Omega\subseteq\mathbb{R}^n$ being "sufficiently regular", for example being the image of a $C^1$ mapping from $\mathbb{R}^p$ for some $p\ge1$. We may then consider the mapping $$ g:\...
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261 views

Lebesgue measure as a fixpoint: change of variables formulas

This question is inspired by several others on a similar topic: see e.g. this one and a sequence of linked questions. Let us so far focus on $\Bbb R^n$ endowed with standard Borel structure. For any ...
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281 views

Why can I integrate something like “infinitesimal part” to calculate the length, area, volume, etc.?

Let me try to elaborate the question by an example: I want to calculate the length of a straight line like $y=x$ between $[0, 1]$. By conventional way I would do a definite integration of integrand $...
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321 views

Does every smooth surjective function have a smooth right inverse?

If you feel this question might be too broad, let me know and I’ll try to get more specific. If $r \colon I → J$ is a smooth surjective function between perfect subspaces $I$ and $J$ of $ℝ$, can ...
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191 views

Algorithm to calculate multiple integral.

One of the major difficulties of student in advanced calculus (including myself when student) is to obtain the extremes of repeated integrals to calculate the volume integral in $R^n$ i.e. transform ...
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87 views

$x\mapsto f(x^{1/p})$ is smooth if and only if $f^{(n)}(0)=0$ whenever $p\nmid n$

This is a slight generalization of something I got stuck on when trying to do Problem 2-5 from Introduction to Smooth Manifolds by John M. Lee (which uses the case $p=3$). Let $p\ge 1$ be an ...
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855 views

“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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650 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 non-...
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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 )\frac{1}{n!}\...
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271 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 Fourier-...
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434 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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248 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 $a^2+b^...
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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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189 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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355 views

Differential Forms

I just had a general question about differential forms. Background. Let $f = f_k$ and define $f^{k-1}$ on $I^{k-1}$ by $$f_{k-1}(x_1, \dots, x_{k-1})= \int_{a_k}^{b_k} f_{k}(x_1, \dots, x_{k-1}, x_k) ...
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193 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 z^{m_m}...
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64 views

If $f$ is defined on $\mathbb{R}$ and $f$ is unbounded, is it necessarily true that $\lim_{|x|\to\infty} |f(x)| = \infty$?

This question comes from the following problem: A real-valued function $f$ defined on $\mathbb{R}$ has the following property: For every positive $\epsilon$, there exists positive $\delta$ such ...
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43 views

Baby Rudin Exercise 4.13 Alternate Proof Verification

I would like to know if my proof of ex 4.13 is correct. Thanks! Exercise 4.13 in Rudin asks: Let $E$ be a dense subset of a metric space $X$, and let $f$ be a uniformly continuous real function ...
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178 views

Is this function increasing?

I'm stuck in showing that the following function is increasing over the domain $\left[0,x_0\right]$: $$\Pi\left(z\right) = \int_{0}^{\phi\left(z\right)}\int_{x}^{\bar{x}}\left(2y-b\left(x\right)-x\...
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Show that if $f(x,t)$ and $f_x(x,t)$ are continuos on $D$ then $F'(x)=\int_c^d f_x(x,t)\mathrm dt$

I want to check a proof. This is about Riemann integration. Show that if $f(x,t)$ and $f_x(x,t)$ are continuos on $D$ then $F'(x)=\int_c^d f_x(x,t)\mathrm dt$ Definitions: $D=\{(x,t):x\in[a,b]...
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28 views

On sharp bounds of some dyadic operators

I am now having interest in finding the sharp bounds of some kinds of dyadic operators which map $L^p(\mathbb{R})$ to $L^p(\mathbb{R})$. For example, the martingale transform $T_{\sigma}$ which ...
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83 views

Why is the unit circle definition of trig functions not rigorous enough?

It has recently come to my attention that the usual unit circle derivation of the elementary trigonometric functions isn't considered rigorous enough. Apparently, this has to do with problems ...
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67 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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57 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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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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39 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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74 views

How close are we to knowing the rate of convergence to $0$ of $\prod_{p\le x}(1-1/p)^{-1}-e^\gamma\log x $?

This is a question related to an earlier one of mine, which I may answer myself eventually, as I have learnt more about the topic. Despite what one can read on the MathWorld page about Mertens' third ...
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85 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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38 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 $H^{s}(\...
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Intuition about the Bernstein polynomials proof of the Weierstrass approximation theorem

The Weierstrass approximation theorem can be stated as follows: Let $f\in C([a,b])$, then there exists a sequence $(p_n)_{n\in \mathbb{N}}$ of polynomials in $[a,b]$ such that $(p_n)$ converges ...
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59 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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75 views

If $f$ and $g$ are analytic functions, does $f \circ g - g \circ f = g- f$ implies $f = g$?

The question is in the title. There are actually two variants, one for real analytic functions and the other one for complex analytic functions. It came up as an attempt to solve this question by ...
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84 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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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:\ \ \int_0^1\|f(...
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210 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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76 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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62 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)\; dy=\frac{1}{...