Questions tagged [real-analysis]

For questions about real analysis, a branch of mathematics dealing with limits, convergence of sequences, construction of the real numbers, the least upper bound property; and related analysis topics, such as continuity, differentiation, and integration through the Fundamental Theorem of Calculus. This tag can also be used for more advanced topics, like measure theory.

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On the Constant Rank Theorem and the Frobenius Theorem for differential equations.

Recently I was reading chapter $4$ (p. $60$) of The Implicit Function Theorem: History, Theorem, and Applications (By Steven George Krantz, Harold R. Parks) on proof's of the equivalence of the ...
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32 votes
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Constructing an infinite chain of subfields of 'hyper' algebraic numbers?

This has now been cross posted to MO. Let $F$ be a subset of $\mathbb{R}$ and let $S_F$ denote the set of values which satisfy some generalized polynomial whose exponents and coefficients are drawn ...
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29 votes
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557 views

The sequence $\,a_n=\lfloor \mathrm{e}^n\rfloor$ contains infinitely many odd and infinitely many even terms

PROBLEM. Show that the sequence $\,a_n=\lfloor \mathrm{e}^n\rfloor$ contains infinitely many odd and infinitely many even terms. It suffices to show that the terms of the sequence $$\,b_n=\mathrm{e}^...
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28 votes
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Witt's proof of Gelfand-Mazur / Ostrowski's theorem

Now asked on MathOverflow. Background: It seems that, after his groundbreaking work on quadratic forms and inventing Witt vectors, Ernst Witt developed the hobby of giving extremely short proofs to ...
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25 votes
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591 views

A subadditive bijection on the positive reals

Question. Does there exist a subadditive bijection $f$ of the positive reals $(0,\infty)$ such that $$ \liminf_{x\to 0^+}f(x)=0 \,\,\,\text{ and }\,\,\,\limsup_{x\to 0^+}f(x)=1\,? $$ Ps1. I guess the ...
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21 votes
0 answers
492 views

A question connected with the decomposition of a functional on $C(X)$ on Riesz and Banach functionals

Let $X$ be a metric space and let $C(X)$ be a family of all bounded and continuous functions from $X$ in $\mathbb{R}$. We call a positive linear functional $\varphi: C(X) \rightarrow \mathbb{R}$ the ...
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20 votes
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339 views

Extension of Vector Field in the $\mathcal{C}^r$ topology.

Let $M\subset \mathbb{R}^n$ be a compact smooth manifold embedded in $\mathbb{R}^n$, we define $$\mathfrak{X}(M) := \{X: M \to \mathbb{R}^n;\ X\mbox{ is smooth and }\ X(p) \in T_p M \subset \mathbb{R}^...
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20 votes
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994 views

Can fundamental theorem of algebra for real polynomials be proven without using complex numbers?

Update 24 Nov 2015: It is solved. Please refer to this arXiv paper. For polynomials with real coefficients, I am trying to prove the following version of fundamental theorem of algebra, which avoids ...
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18 votes
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319 views

Show that there are no real solutions of $1 + \sum_{n = 1}^{\infty} \frac{x^n}{\prod_{k=1}^{n} H_k} = 0$ where $H_k = \sum_{i=1}^{k} \frac{1}{i}.$

[Edited] Show that there are no real solutions of $$1 + \sum_{n = 1}^{\infty} \frac{x^n}{\prod_{k=1}^{n} H_k} = 0$$ where $$H_k = \sum_{i=1}^{k} \frac{1}{i}.$$ I managed to prove this, and I want to ...
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18 votes
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Find $f$ such that $f \star f(x) = \frac{1}{1-x}$.

I'm looking for a measurable function $f$ defined on $]0,1[$ such that : $$f \star f(x) = \int_{0}^1 f(x-y) f(y) \ \mathrm{d}y = \frac{1}{1-x}$$ for (almost) any $x \in ]0,1[$. Is it possible to find ...
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18 votes
1 answer
260 views

Supremum of $\int_{-\infty}^{\infty}\frac{p'(x)^2}{p(x)^2+p'(x)^2}\,\mathrm{d}x$

Question. Let $P_d = \{ p \in \mathbb{R}[x] : \deg p = d\}$ denote the set of all degree $d$ polynomials with real coefficients. Also, for $p \in \mathbb{R}[x]$, define $$ I(p) = \frac{1}{\pi} \int_{-\...
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18 votes
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498 views

Stokes' Theorem general case

With the following lemma : Lemma : Let $f_{+},f_{-} : U \longmapsto \mathbb{R}$ be $C^{1}$ maps with $f_{-} \leq 0 \leq f_{+}$ with $U \subset \mathbb{R}^{n}$ open and bounded with $C^{1}$ boundary. ...
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18 votes
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911 views

Refinement of a famous inequality

I refine a famous inequality this is the following : Let $x,y>0$ then we have : $$x^n+y^n\leq \Big(\frac{x^n+y^n}{x^{n-1}+y^{n-1}} \Big)^n+\Big(\frac{x+y}{2}\Big)^n$$ It's equivalent to : $$x^n+1\...
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18 votes
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A closed $1$-form on a convex open set is exact

Baby Rudin Exercise 10.24: Let $\omega = \sum a_i(\mathbf x) \, dx_i$ be a $1$-form of class $\mathscr{C}''$ in a convex open set $E \subset \mathbb{R}^n$. Assume $d \omega = 0$ and show that $\...
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17 votes
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Lagrange multipliers in Calculus of Variations

I am trying to learn about Calculus of Variations and I am beginning to see some constrained optimization problems in the domain of functionals, by using Lagrange multipliers. It seems that things ...
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16 votes
0 answers
574 views

Is the minimum of this functional $C^{\infty}$?

The problem: Let us define $$ \mathscr{F}(u)=\int_0^1(u'(x))^4-e^x\sin (u(x))\, \mathrm{d}x $$ for $u \in W^{1,1}([0,1])$ such that $u(0)=A$ and $u(1)=B$. It don't matters what $A$ or $B$ are, because ...
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16 votes
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Theorem 6.17 in Baby Rudin: $\int_a^b f \,d\alpha = \int_a^b f(x) \alpha^\prime(x) \,dx$

Here is Theorem 6.17 in the book Principles of Mathematical Analysis by Walter Rudin, 3rd edition: Assume $\alpha$ increases monotonically and $\alpha^\prime \in \mathscr{R}$ on $[a, b]$. Let $f$ ...
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16 votes
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Is there a book only about epsilon delta proofs?

I want to know if there is such book, with beautiful epsilon delta proofs of all kind.
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15 votes
1 answer
572 views

About the inequality : $x^{x^{x^{x^{x^x}}}}\geq x^{x^{x^{((e-2)(1+e))x\left(1+\sqrt{x}\left((\sqrt{x})^3-1\right)\right)}}}\geq x^{x^{\frac{16}{27}}}$

This inequality is due to user RiverLi : Let $0<x\leq 1$ then we have : $$x^{x^{x^{x^{x^x}}}}\geq x^{x^{\frac{16}{27}}} \geq 0.5x^2+0.5$$ I propose another one wich states : Let $0<x\leq 1$ ...
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15 votes
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629 views

Bounding a polynomial from below

Let $\sigma >0$ be fixed. For even $k \in \mathbb{N} \cup \{0\}$, we consider the polynomial \begin{equation} \varphi_k(x) = \sum_{j=0}^{k} (-1)^j {k \choose j} b_j \, x^{2j} \quad x \in (-1,1), \...
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15 votes
0 answers
465 views

Increasing derivatives of recursively defined polynomials

Consider recursively defined polynomials $f_0(x) = x$ and $f_{n+1}(x) = f_n(x) - f_n'(x) x (1-x)$. These polynomials have some special properties, for example $f_n(0) = 0$, $f_n(1) = 1$, and all $n+1$...
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15 votes
0 answers
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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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14 votes
0 answers
103 views

How do permutations of $\Bbb N$ affect series?

Let $G$ be the group of all permutations of $\mathbb{N}$ and $\sum a_n$ a conditionally convergent series of reals. What do we know about how $G$ "acts" on this series? We can partition $G$ ...
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14 votes
0 answers
897 views

Definitions of Measurability: Outer-inner measure convergence vs. Caratheodory criterion

If we are looking over subsets of $\mathbb R$ and considering the outer measure defined exactly as $$\mu^*(A) = \inf\left\{ \sum_{k=1}^\infty \ell(I_k) \text{ where the $I_k$ are open intervals such ...
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14 votes
0 answers
408 views

Combinatorial Proofs of Real Analysis Identity

In this question, a proof using real analysis is given of the following identity: $$ \sum_{n=1}^{\infty} \frac{(n-1)!}{n \prod_{i=1}^{n} (a+i)} = \sum_{k=1}^{\infty} \frac{1}{(a+k)^2}$$ Is there a ...
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14 votes
1 answer
359 views

Approximating the area below average of a concave function

Given a non-decreasing concave function $f:[0,1]\rightarrow \mathbb{R}^+$. Define \begin{align*} F(n)=\displaystyle\sum_{i=1}^n \min\left\{\frac{1}{n},\frac{f\left(\frac{i}{n+1}\right)}{n+1}\right\} ...
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0 answers
386 views
+100

Does there exist $f:\Bbb{R}\to \Bbb{R}$ additive onto function such that $f(F) \subset \Bbb{R}$ has the property of Baire for every $F$?

Let $F\subset\Bbb{R} $ intersect every uncountable $\mathcal{F}_{\sigma}$ set. $B\subset \Bbb{R}$ is said to have the property of Baire if $B=U\triangle M$ where $U$ is open and $M$ is meager. Does ...
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  • 9,255
13 votes
1 answer
557 views

Function of two sets

Let $U$ be the set of all nonempty subsets of $[0,1]$ that are a union of finitely many closed intervals (where an "interval" that is a single point does not count as an interval). Does ...
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13 votes
0 answers
169 views

Symbolic approximation through integration by parts

This is a slightly soft question. Suppose I have an integral $f(x) =\int_a^x g(t) dt $ which cannot be expressed in terms of elementary functions. One might still be able to integrate by parts to get ...
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  • 2,564
13 votes
0 answers
547 views

How to prove this lemma related to Rolle's theorem

For any function $f$ denote by $Z(f)$ and $Z_o(f)$ the cardinalities of $f^{-1}(0)\cap[0,1]$ and $f^{-1}(0)\cap(0,1)$, respectively. Let $H=\{f\in C^\infty(\mathbb{R}): \text{supp}(f) = [0,1]\}$ From ...
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  • 1,184
13 votes
1 answer
1k views

Intuition about the Bernstein polynomials proof of the Weierstrass approximation theorem

The Weierstrass approximation theorem can be stated as follows: Let $f\in C([a,b])$. There exists a sequence $(p_n)_{n\in \mathbb{N}}$ of polynomials in $[a,b]$ such that $(p_n)$ converges uniformly ...
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  • 24.4k
13 votes
0 answers
486 views

How to construct examples of functions in the Spaces of type $\mathcal{S}$

There are $3$ $\mathcal{S}$-type Spaces, namely $\mathcal{S}_\alpha\:,\: \mathcal{S}^\beta\:,\:\mathcal{S}_\alpha^\beta$. They are defined by: $\mathcal{S}_\alpha: |x^k\varphi^{(q)}(x)|\le C_qA^kk^{...
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  • 3,405
13 votes
1 answer
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Heat Kernel Property

Let $\phi$ be the Heat Kernel in $\mathbb{R}^n$, i. e. $$\phi (x,t)={(4\pi t)}^{-n/2}\exp\left( - \frac{\mid x \mid ^2}{4t}\right)$$ and let $u$ satisfy Heat equation. Show that: $$\frac{d}{dt}\int\...
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13 votes
0 answers
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Question regarding the Kolmogorov-Riesz theorem on relatively compact subsets of $L^p(\Omega)$.

Usually, the Kolmogorov-Riesz theorem is quoted for $L^p(\mathbb R^n)$, but I am looking for versions considering spaces over subsets in $\mathbb R^n$. The following is from the book "Sobolev spaces" ...
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  • 357
12 votes
0 answers
153 views

How close can polynomials be in magnitude?

If $p$ and $q$ are polynomials and $|p(t)|=|q(t)|$ on the interval $[0,1]$, then in fact $p(t)=\omega q(t)$ for some constant phase $\omega$. I am curious about quantitative strengthenings of this ...
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12 votes
0 answers
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Let $f: A \to \mathbb{R^n}$ be of class $C^r$; $Df(x)$ is non-singular for $x\in A$. Show that even if $f$ is not 1-1, the set $B=f(A)$ is open.

Let $A$ be open in $\mathbb{R^n}$; let $f: A \to \mathbb{R^n}$ be of class $C^r$; assume $Df(x)$ is non-singular for $x\in A$. Show that even if $f$ is not one-to-one on $A$, the set $B=f(A)$ is open ...
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11 votes
0 answers
164 views

Rearranging series and "placid" permutations

This question came out of a conversation with my students about Riemann's rearrangement theorem, and the general problem of which permutations are "safe" w/r/t summing infinite series. Let $...
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11 votes
0 answers
116 views

Does there exist any $p >0$ such that $\frac{1}{n^p \sin(n)} \to 0 \;,n\to+\infty$?

Does there exist any $p >0$ such that \begin{equation*} \frac{1}{n^p \sin(n)} \to 0 \;,n\to+\infty \;? \end{equation*} If there is one, what's the infimum of those $p$? Is it also a minimum? I ...
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11 votes
0 answers
141 views

Putnam 2018 - Exercise A.5 - proof check

The problem statement is as follows. Let $f:\Bbb R \to \Bbb R$ be an infinitely differentiable function satisfying $f(0) = 0$ and $f(1) = 1$, and $f(x) \geq 0$ for all $x \in \Bbb R$. Show that there ...
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  • 5,484
11 votes
0 answers
285 views

Can we further refine $\int_{0}^{1}x^{x^x} \ dx=\frac 1 2+\sum_{n=1}^{\infty}(-1)^n\sum_{k=1}^{\infty}\frac {(n-k)^k}{(k+1)^{n+1}} \binom {n}{k}$

Question Can we further refine the integral $$\int_{0}^{1}x^{x^x}\ dx=\frac 1 2+\sum_{n=1}^{\infty}(-1)^n\sum_{k=1}^{n}\frac {(n-k)^k}{(k+1)^{n+1}} \binom {n}{k}$$ ? To compute the result, first ...
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  • 1,384
11 votes
0 answers
276 views

Convergence of a Stochastic Process - Am I missing something obvious?

In the paper On the Convergence of Stochastic Iterative Dynamic Programming Algorithms (Jaakkola et al. 1994) the authors claim that a statement is "easy to show". Now I am starting to suspect that it ...
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  • 1,765
11 votes
0 answers
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Rigorous Proof of Slutsky's Theorem

I was hoping to type up my proof of Slutsky's Theorem and get confirmation on the excruciating details being all correct... Statement of Slutsky's Theorem: $$\text{Let }X_n, \ X,\ Y_n,\ Y,\text{ share ...
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  • 501
11 votes
1 answer
341 views

A Question on certain Hilbert space of continuous functions, and a characteristic of convergence in it

Define $T^k(\Omega)$, $\Omega$ an open subset of $\mathbb{R}^m$ (with a smooth boundary), as a space of function equivalance classes, with the norm defined as $$ \|f\|_{T^k(\Omega)}^2 = \|f\|_{L^2(...
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  • 3,929
11 votes
0 answers
774 views

Schilling's proof of the Feynman-Kac Formula for Brownian motion

This is part of a proof to the Feynman-Kac formula from Schilling's Brownian motion. I need some help understanding the proof to this theorem. Theorem (Kac 1949). Let $(B_t)_{t\ge 0}$ be a $d$-...
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11 votes
0 answers
1k views

Theorem 6.12 (a) in Baby Rudin: $\int_a^b \left( f_1 + f_2 \right) d \alpha=\int_a^b f_1 d \alpha + \int_a^b f_2 d \alpha$

Here is part (a) of Theorem 6.12 in the book Principles of Mathematical Analysis by Walter Rudin, 3rd edition: If $f_1 \in \mathscr{R}(\alpha)$ and $f_2 \in \mathscr{R}(\alpha)$, then $$f_1 + ...
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11 votes
0 answers
265 views

Covering a set $A \subset \Bbb R$ by two families of disjoint intervals taken from given intervals

Let $A \subset \Bbb R$ be a bounded set. Every element $a \in A$ is the center of some given open interval, let's denote it by $I_a=(a-r_a, a+r_a)$. I'm interested in knowing the following: Can we ...
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  • 3,396
11 votes
0 answers
458 views

A very useful lemma for Henstock-Stieltjes integration

I'd like to see a proof (or hints and outlines) for the following lemma, which is very useful to prove some interesting properties, including an Integration by Parts theorem for Henstock-Stieltjes ...
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11 votes
0 answers
530 views

Fréchet L-Spaces

NOTE: The question has now been posted on MathOverflow: Fréchet L-Spaces According to the paper The emergence of open sets, closed sets, and limit points in analysis and topology famous ...
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11 votes
0 answers
2k views

A random variable is symmetric if and only if its characteristic function is real-valued

Quick summary: I am stuck on the implication: $\phi_X$ real-valued $\rightarrow$ $X$ symmetric. Assume you have a probability space $(\Omega, \mathcal{F},P)$, and a random varaiable $X: \Omega \...
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  • 9,458
11 votes
0 answers
694 views

norm of a singular integral operator

My question is from Harmonic Analysis, about the study of singular kernels (in the Calderon Zygmund sense.) Suppose that a kernel $K$ is a singular kernel, extending to a bounded operator on $L^p$, ...
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