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, integration through the fundamental theorem of calculus.

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$f \in L^1(- \infty , + \infty)$ , Find the limit $ \lim_{n \to \infty} \int^{{+ \infty}}_{- \infty} \frac{f(x)e^{nx}}{1+e^{nx}} dx $

$f \in L^1(- \infty , + \infty)$ , Find the limit $ \lim_{n \to \infty} \int^{{+ \infty}}_{- \infty} \frac{f(x)e^{nx}}{1+e^{nx}} dx $ My Attempt $$ \lim_{n \to \infty} \int^{{+ \infty}}_{- \infty} ...
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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 $H$, : ...
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108 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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81 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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136 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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112 views

Operator norm of a convolution

Consider the operator on $L^2(\Bbb R)$, $f\rightarrow f*g$, where $g\geq 0$ is some $L^1$ function. Show the operator is a bounded linear operator with operator norm equal to $||g||_1$. Showing ...
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164 views

If a Radon measure is a tempered distribution, does it integrate all Schwartz functions?

The question might at first sight sound like the answer is trivially "yes", so let me clarify the question a bit. Consider given a Radon measure $\mu$ on $\mathbb{R}^n$. Let ...
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560 views

Proof that a sequence of continuous functions $(f_n)$ cannot converge pointwise to $1_\mathbb{Q}$ on $[0,1]$

As a homework question, we got asked the following: Construct a function $f:[0,1] \rightarrow \mathbb{R}$ which is not the pointwise limit of any sequence of continuous functions Thinking about ...
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134 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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155 views

Dense uncountable proper subgroup of $(\mathbb{R},+)$

Probably someone had asked this question on StackExchange, but can one construct a dense uncountable proper subgroup of $(\mathbb{R},+)$?
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How to find a homeomorphism $\mathbb{R}^{n}\rightarrow\mathbb{R}^{n}$ having certain properties

Let $n\ge 2$ and let $C$ be a cantor space in $\mathbb{R}^{n}$. That is, $C$ is homeomorphic to the cantor ternary set. Let $x$ and $y$ be two points in $\mathbb{R}^{n}-C$, and let $L_{xy}$ be the ...
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132 views

Equivalence of the Lebesgue integral and the Henstock–Kurzweil integral on nonnegative real functions

Let $f:[a,b]\to[0,\infty)$ ($\mathbb{R}\ni a<b\in\mathbb{R}$), and fix $c\geq0$. I want to establish the equivalence of the concepts of Lebesgue integrability and Henstock–Kurzweil integrability ...
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157 views

Simple problem of a differentiable function

Please, can somebody help me with this problem? I tried to use the Mean Value Theorem, but couldn't solve it. Let $g: [a,b]\rightarrow\mathbb{R}$ a differentiable function on $[a,b]$. If ...
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135 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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131 views

Is there a subsequence of $a_n = n \sin(n)$ which tends to $0$?

I know there is such a subsequence for $b_n = \sin(n)$. What about $a_n = n\sin(n)$?
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187 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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73 views

$M$ is compact, non-empty, perfect, and $M \cong M \times M$. Must $M$ be homeomorphic to the Cantor set, the Hilbert cube, or some combination?

Assume that $M$ is compact, non-empty, perfect, and homeomorphic to its Cartesian square, $M \cong M \times M$. Must $M$ be homeomorphic to the Cantor set, the Hilbert cube, or some combination of ...
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79 views

Inverse of a differentiable function equal to its derivative then f is analytic

I've found a nice problem concerning analytic functions. Here it is: Let $f: (0, \infty) \rightarrow \mathbb{R}$ be a function differentiable on $(0, \infty)$ and such that $f^{-1} = f'$. Prove that ...
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143 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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159 views

Proving that a set is closed in $L^2(\mathbb{R})$

this is my first question here so i hope i don't do anything wrong. Excuse any spelling or grammar mistakes, english isn't my mother tongue. I'm reading this paper for my bachelor thesis and have ...
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369 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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68 views

Smoothness in $\mathbb{R}^n$

Embarrasingly simple question, but I got the feeling that I cannot see the forrest for the trees right now: If I have a function $f: \mathbb{R}^2 \rightarrow \mathbb{R}$ and want to show that it is ...
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329 views

Equality condition in Minkowski's inequality for $L^{\infty}$

I am trying to find out when equality holds in Minkowski's inequality for $L^{\infty}$ (i.e. a necessary and sufficient condition for equality). I did a search and there was a discussion for the case ...
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103 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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591 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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236 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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208 views

Is there a continuous nowhere Lanczos generalized differentiable function?

The Lanczos generalized derivative comes from local regression, and is described in this pdf. (You do not need to read that to understand my questions.) If you do read it, observe that I'm pulling ...
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177 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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229 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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139 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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174 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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199 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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171 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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Two question about how to compute this integral limit

Let $f: (-\pi,\pi]\to \mathbb R$ be continuous and let $p_x (u) = {(f(u+x) - f(x)) \cos ({u \over 2}) \over \sin ({u \over 2}) }$. I want to show that $$ \int_{-\pi}^\pi p_x(u) \sin (Nu) du \to 0$$ ...
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If $f_n(t):=f(t^n)$ converges uniformly to continuous function then $f$ constant

Let $f \colon [0,1] \to \mathbb R$ be continuous and let $f_n$ be defined by $$ f_n(x):=f(x^n), \qquad x \in [0,1], \,\, n \in \mathbb N. $$ Suppose there exists a continuous function $g$ on ...
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Functions with compact support

I have a question about a convergence of functions with compact support. SETTING Let $d\geq 3$ and $U \subset \mathbb{R^{d}}$ be open and $dx$= Lebesgue measure on $U$. Let $b_{i},c,d_{i} \in ...
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Does this arithmetic operation have a name

I've came across the following product-like operation on reals $$ a\times b:=1 - (1-a)(1-b). $$ This operation is commutative, associative and has $1$ as a zero element and $0$ as a unit element: $$ ...
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Jordan decomposition of linear functionals

Let $X$ be a locally compact Hausdorff space. Also, let $C_0(X,\mathbb R)$ denote the vector space of such continuous functions $f:X\to\mathbb R$ that the set $\{x\in X\,|\,|f(x)|\geq\varepsilon\}$ is ...
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A formula similar to $\int_a^bf(x)dx=\mu\left[a,b \right]$ for $f^p$.

Let $\mu$ be an absolutely continuous measure with respect to the Lebesgue measure on $\mathbb{R}$ , and $f:\mathbb{R}\to \mathbb{R^+}$ its Radon-Nikodym derivative . We can write $\int_a^bf(x)dx$ in ...
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Clarification on tetration

So far when I looked at tetration I noticed it had a recursive relation. It's $t_2=2^{(t_1)}.$ For example if we start at point $(0,1)$, we can take the x-value of $0$, and $2^0=1$, then we take $1$ ...
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Analysis or (abstract) algebra first?

Which one would you recommend? I only know calculus and linear algebra when it comes to university-level mathematics. Is one required to understand the other?
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Proving u-substitution the hard way — use only definition of integration with partitions

I am trying to prove integration by substitution, i.e. for $f:[c,d] \rightarrow \mathbb{R}$ continuous and $\phi: [c,d] \rightarrow [a,b]$ continuous on $[c,d]$ and $C^1$ on $(c,d)$. Then define ...
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Weak topology on $L^p,~p> 1$

How looks like the weak topology in the particular case $X=L^p$, I mean, is possible to detail this topology beyond standar form: Arbitrary union of finite intersections open pre-images of opens ...
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Evaluation of the double integral $\iint_{[0,1] \times [0,1]}{\left(\frac{xy(1-x)(1-y)}{2-xy}\right)^n \frac{dxdy}{2-xy}}$

How to prove that $$\large \lim_{n \to \infty} \left(\iint_{[0,1] \times [0,1]}{\left(\frac{xy(1-x)(1-y)}{2-xy}\right)^n \cdot\frac{1}{2-xy}\; ...
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Integral $I=\int_0^1 \frac{\log x \log (1+x) \log(1-x) \log(1+x^2)\log(1-x^2)}{x^{3/2}}dx$

Hi I am trying to integrate and obtain a closed form result for $$ I:=\int_0^1 \frac{\log x \log (1+x) \log(1-x) \log(1+x^2)\log(1-x^2)}{x^{3/2}}dx. $$ Here is what I tried (but I do not think this is ...
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Convergence of $\sum_n |\frac{\cos(3^n)}{n}|$

So a recent post asked about convergence of $\sum_n |\frac{\cos(2^n)}{n}|$, and using double-angle formula for $\cos$ it could be shown that for each pair of consecutive terms, at least one term had ...
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74 views

Rudin's Principle of Mathematical Analysis Problem

If ${(s_n)}$ is a complex sequence, define its arithmetic means $\sigma_n$ by $$\sigma_n = \frac{s_0 +s_1 + ... + s_n}{n+1}$$ Assume that $M < \infty, |na_n| \leq M$ for all n and lim ...
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Is there a measure on $\mathbb{R^3}$ other than volume?

Of course there is the trivial measure where each subset of $\mathbb{R^3}$ have measure zero. But I am asking for a measure $\mu$ for which $\mu(E)=1$ where $E=[0,1]^3$. I thought about the radius, ...
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62 views

Showing that $\sum_{n=1}^\infty \frac{x}{n(1+nx^2)}$ Converges Uniformly via the Weierstrass M-test

Setting: Let $f_n(x):\mathbb{R} \rightarrow \mathbb{R}$ s.t. $$ f_n(x) = \frac{x}{n(1+nx^2)} $$ and let $f$ denote the series of the $\{f_n\}$: $$ f(x) = \sum_{n=1}^\infty f_n = \sum_{n=1}^\infty ...
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243 views

Show that the iterated $\ln^{[n]}$ of tetration(x,n) is nowhere analytic

$$f(x) = \lim_{n\to \infty} \ln^{[n]} x \uparrow\uparrow n$$ The conjecture is that $f(x)$ is monotonic and infinitely differentiable at the real axis, but nowhere analytic; because at each point on ...