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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7
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249 views

Convergence of indicator functions in $L^2[0,1]$ when $m(\limsup(E_n)\setminus \liminf(E_n)) = 0$

I am trying to solve a qualifying exam problem. I would like to know what can be said about the convergence of the indicator functions $I_{E_k}$ in $L^2[0, 1]$ when it's known that $m(\limsup ...
7
votes
0answers
623 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 ...
6
votes
0answers
26 views

A version of Ampère's law

The most common proof that I have found of the fact that Ampère's law is entailed by the Biot-Savart law uses the fact that, if $\boldsymbol{J}:\mathbb{R}^3\to\mathbb{R}^3$, $\boldsymbol{J}\in ...
6
votes
0answers
138 views
+50

Which properties characterize $\sin, \cos$?

I know a few properties of $\sin$ and $\cos$, for example: $\sin^2+\cos^2=1$ $\sin (a+b) = \sin a\cos b+\cos a\sin b$. $\cos (a+b) = \cos a\cos b-\sin a\sin b$. $\sin (x+\delta) = \sin x$ for some ...
6
votes
0answers
124 views

How do people pick $\delta$ so fast in $\epsilon-\delta$ proofs

For example, in a proof that shows $f(x) = \sqrt(x)$ is uniformly continuous on the positive real line, the proof goes like: Let $\epsilon > 0$ be given, and $\delta = \epsilon^2$.... Or to ...
6
votes
0answers
42 views

boundness of linear map

Let $X$ and $Y$ be Banach spaces, $T : X\rightarrow Y$ a linear map such that $f \circ T\in X^*$ for every $f\in Y^*$, then $T$ is bounded. I've tried to show that $T$ is closed, but I think that's ...
6
votes
0answers
165 views

How do i evaluate this sum $\sum\limits_{n=1}^{\infty} \frac{(-1)^{n+1}}{n^2n!}$?

How do I evaluate this sum: $$\sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n^2n!}$$ Note: The series converges by the ratio test. I have tried to use this sum:$$ \sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n}= ...
6
votes
0answers
97 views

Let $(X,\Sigma, \mu)$ be a measure space. If $f$ is integrable and $\int_E {f}\,{d\mu} = 0$ for all $E\in\Sigma$ then $f=0$ almost everywhere.

My attempt: I called: $A = \{ x\in X : f(x) \ne 0 \}$ $B_n = \{ x\in X: f(x) \gt \frac{1}{n} \}$ $C_n = \{ x\in X:f(x) \lt -\frac{1}{n} \}$ $B = \bigcup_{n=1}^{\infty}{B_n}$ $C = ...
6
votes
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82 views

Approximating $x^m$ by exponentials

What's a nice explicit example of a sequence of functions $f_n$: $\mathbb{R}\longrightarrow\mathbb{R}$ of the form $$a_1e^{c_1x}+\ldots + a_ke^{c_kx},\;\;\; a_i, c_i\in\mathbb{R}$$ that converges to ...
6
votes
0answers
155 views

Transexponential Functions

Recall that $\exp(1,x) = e^x$ and $\exp(n+1,x) = e^{\exp(n,x)}$. Recall that $f(x)$ is transexponential if $f(x)$ is eventually greater than $\exp(n,x)$ $\forall n \in \mathbb{N}$ I am looking for a ...
6
votes
0answers
170 views

Anti-random reals

EDIT: This has now been crossposted at MO: http://mathoverflow.net/questions/219366/antirandom-reals. This is partially motivated by my question at mathoverflow: ...
6
votes
0answers
88 views

Every $f\in\omega^\omega$ is bounded by the “increasing enumeration” of the intersection of a countable dense set and a dense open set in $\mathbb{R}$

I am studying the theorem 2.2.6 of "On the structure of the real line" of book Bartosznky-Judah. In the proof of theorem 2.2.6 the part $(4) \to (5)$ $(4)$ for every family of dense open subsets ...
6
votes
0answers
177 views

compare norms on $\mathcal{B}(H)$

Given a Hilbert space $H$ and $a$ be a real numbers $\geq‎‎‎ 1$ , let $S_1(H)$ denote the space of trace-class operators on $H$, with the trace-class norm or Schatten 1-norm. That is $$ \Vert T ...
6
votes
0answers
120 views

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 ...
6
votes
0answers
144 views

Proving the function $ f $ is continuous on $ [0,1] $

I'm trying to prove that the following function $ f $ is continuous on $ [0,1] $. The function $ f:[0,1]\rightarrow [0,1] $ is defined as follows. Let $ x\in [0,1] $. Then $ x= \sum\limits_{n = ...
6
votes
0answers
84 views

Estimating the (double) Riesz transform.

I'm trying to verify the following estimate, which appears in a paper I'm reading. It seems I'm missing something easy, I just can't figure this out. $\textbf{Background}:$ For a function $f \in ...
6
votes
0answers
140 views

The dual of the Banach space $C(\Omega)$

It is well-known that the dual of the Banach space $C([0,1])$, i.e. the space of all continuous functions on the interval, is the space of all functions of bounded variation on the interval, ...
6
votes
0answers
87 views

Dimension score for metric spaces $\mathbb{R}^n$

Given any metric space $(X,d)$, define its score $S(X)$ to be the smallest value of $k$ such that for every $x\in X$ and $r>0$, the ball $B(x,r)$ is covered by at most $2^k$ balls of radius $r/2$. ...
6
votes
0answers
139 views

Finitely additive function on an infinite set, s.t., $m(A)=0$ for any finite set and $m(X)=1$ (constructive approach)

Other exercise which I found in Dudley's Analysis book: Show that there is a measure on a infinite set $X$, defined on $2^X$ s.t. is finitely additive, $m(A)=0$ for any finite set and $m(X)=1$. ...
6
votes
0answers
108 views

Generating Function for Associated Stirling Numbers $b(n,k)$

I am trying to identify or find the ordinary generating function (not the exponential generating function) for the Associated Stirling numbers of the Second kind, denoted $$b(1;n,k)=b(n,k)$$ These ...
6
votes
0answers
135 views

Proposed proof of analysis result

Hi please advise on my proof of the following result: Assume that $I \subset \mathbb{R}^{n}$ is convex, bounded open set with Lipschitz boundary and let $u_{m},u$ be such that $$u_{m} ...
6
votes
0answers
59 views

Reference for the fact that a smooth function analytic on every line is itself analytic

Let $f \in \mathcal C^\infty(\mathbb R^p)$ ($p \geq 2$) be a smooth function such that the functions $g_d(t) := f(td)$ are all analytic for all $t \in \mathbb R$ and all $d \in \mathbb R^p.$ (i.e. $f$ ...
6
votes
0answers
101 views

Is $\sigma$-finiteness really a necessary condition for this problem?

Question: Let $(X, \mathcal A, \mu)$ be a measure space and suppose $\mu$ is $\sigma$-finite. Suppose $f$ is integrable. Prove that given any $\varepsilon$, there exists a $\delta >0$ such that ...
6
votes
0answers
262 views

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 ...
6
votes
0answers
148 views

Uniqueness solutions of $dx/dt = f^2(x) + e^{-t}$.

Someone can help me in the following problem? Is a question of Zhang. Let $f(x)$ be continuous for $x \in \mathbb{R}$, show that $dx/dt = f^2(x) + e^{-t}$ has the property of uniqueness of ...
6
votes
0answers
102 views

Asymptotic property of integral involving Bessel function.

Consider the following integral $$ I(s)=\int_{0}^{\infty}{J_{\frac{n-2}{2}}}(sr)r^{A+1}(e^{-r^{2\alpha}}-1)dr, $$ where $J_{\frac{n-2}{2}}$ is the Bessel function of order ${\frac{n-2}{2}}$, $s, A, ...
6
votes
0answers
103 views

$f \in L^1(- \infty , + \infty)$ , 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} ...
6
votes
0answers
136 views

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 ...
6
votes
0answers
182 views

Sequence of convex functions converges uniformly

I am working on the following problem. Let $f_{n}: [a, b] \rightarrow \mathbb{R}$ be a sequence of convex functions. Furthermore, for each fixed $x \in [a, b]$, suppose $f(x) = \lim_{n ...
6
votes
0answers
204 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 $a_n = n\sin(n)$?
6
votes
0answers
271 views

Behaviour at infinity of a function in terms of first and second derivatives

In a paper (dealing with spectra of certain Schrodinger operators) I found the following assumption for a function $f\in C^\infty(\mathbb R^n;\mathbb R)$: there exists a constant $C>0$ and a ...
6
votes
0answers
169 views

Does $\displaystyle \lim_{m \to +\infty}f_{2,m}(x)$ converge?

This is related to a previous question where, as stated there, $f_{2}(n)$ gives the greatest power of $2$ that divides $n$. Specifically the sequence $\lbrace ...
6
votes
0answers
235 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 ...
6
votes
0answers
234 views

Characterize the continuous functions with finite right-hand derivative for at least one point of $[0, 1]$

Let $(E,d_\infty)$ the metric space of continuous functions defined on $ [0,1] $, with $$ d_\infty(f,g)=\sup\{ |f(x) - g(x)| : x\in [0,1] \}. $$ For all $ n\in \mathbb{N} $ let $$ F_n = \{ f: \exists ...
5
votes
0answers
70 views

Optimization of approximate functions using varying objective function

Let $g(\theta;x)$ and $f(\theta;x)$ be two convex functions such that $g$ asymptotically approximates $f$: $g(\theta;x)\approx f(\theta;x)$, specifically: $$ |g(\theta;x)-f(\theta;x)| \leq ...
5
votes
0answers
58 views

Capacity vs measure of a set - intuitive understanding

There is a concept of measure of "largeness" of a set, called capacity. The intuition is, instead of physical largeness (measured by Hausdorff or Lebesgue measure), capacity measures how good a given ...
5
votes
0answers
134 views

Old & cool integral $\int_0^{\pi} \sin^{b-1}(x) \sin(a x) \ dx=\frac{\pi \sin(a \pi/2)}{2^{b-1}b B\left(\frac{b+a+1}{2},\frac{b-a+1}{2}\right)}$

Here is an integral that appears in the table of integrals by Gradshtein and Ryzhik, it was also studied by Ramanujan (not sure his original solution was found - it seems it doesn't appear in any of ...
5
votes
0answers
77 views

$f$ and its $n$-th derivative are bounded imply the $i$-th derivative is bound for $1\leq i\leq n-1$

Let $f:\mathbb{R}\to \mathbb{R}$ be a $n$ times continuously differentiable function such that $f$ and $f^{(n)}$ are bounded. Show that there is a constant $C$ such that ...
5
votes
0answers
58 views

Estimation of a sequence related to the Stirling's formula

I need to show that $$n!=\left(\frac{n}{e}\right)^n\sqrt{2\pi n}e^{\lambda_n}$$ where $$\frac{1}{12(n+1)}<\lambda_n$$ I calculated that $$\lambda_n=\ln n!+n-n\ln n -\frac{1}{2}\ln(2\pi n)$$ On ...
5
votes
0answers
45 views

Tail estimate for $L^1$ functions.

Suppose $f\in{L^1(\mu)}$ for some probability measure $\mu$. Pick $\epsilon>0$ and let $A_n=\{x:|f(x)|>\epsilon{n}\}$. I want to show that $$\mu(A_1)+\mu(A_2)+\dots<\infty$$ My first ...
5
votes
0answers
137 views

An application of intermediate value theorem to prove the existence of solutions

Let $f$ be a continuous function on $[0,1]$ with $f(0)=f(1)=0$. Prove that, for each $\alpha \in (0,1)$, there exists $x_1, x_2\in (0,1)$ such that $$f(x_1)=f(x_2)$$ with $x_1-x_2=\alpha$ or ...
5
votes
0answers
296 views

Cyclic vector in $\ell^2(\mathbb{Z})$ space

Suppose we look at $\ell^2(\mathbb{Z})$, which contains vectors $c=(\dots,c_{-2},c_{-1},c_0,c_1,\dots)$ with $\sum|c(n)|^2<\infty$. Define the right-translation operator by $$R:\{c(n)\}\mapsto ...
5
votes
0answers
34 views

On a step in Egorov's measure theorem

I am in the midst of proving the following Theorem Let $(f_n)$ be a sequence of measurable functions converging almost everywhere to a function $f$ on a measurable set $E\subset [0,1]=\Omega$. Then ...
5
votes
0answers
49 views

Show that if $f$ is measurable then $\{x\in A:c=f(x)\}$ is measurable for each $c$.

Let $A$ be a bounded measurable subset of $\mathbb{R}$. Show that if $f:A\rightarrow \mathbb{R}$ is measurable then $\{x\in A:c=f(x)\}$ is measurable for each $c$. Choose real $c$. Since $f$ is ...
5
votes
0answers
94 views

$\int f_n g \, d\mu \to \int fg \, d\mu$ for all $g$ which belongs to $\mathscr{L}^q (X)$ (exercise)

Hi everyone I find the following exercise and I'd like to know if my answer is correct. Let $(X, \mathscr A, \mu)$ a finite measure space. Let $\{f_n\}$ a sequence of measurable functions such ...
5
votes
0answers
183 views

Alternative Proof of $\sqrt{2}$ is irrational

Can anyone check if this proof is correct. Thank you. Proof that $\sqrt{2}$ is irrational. Let x = $\sqrt{2}$ then $x^2=2$ and $x^2-2=0$ By the Rational Root Theorem, we have: the number $1$ ...
5
votes
0answers
47 views

Definitions/ intuition for differential forms

I've read about differential forms in the Princeton Companion to Mathematics by Gower and in baby Rudin and I'm having trouble reconciling the two expositions. Rudin says a k-form in $E$ ($\subset ...
5
votes
0answers
89 views

Does this inequality hold? Proof / Counterexample

Does the following inequality $ \int_0 ^\infty x^2 |\frac{d}{dx}f(x)|^2 dx - \int_0 ^\infty x |f(x)|^2 dx + 2\pi (\int_0 ^\infty x^2 |f(x)|^2 dx) (\int_0 ^\infty x |f(x)|^2 dx) > -\frac{1}{8\pi} ...
5
votes
0answers
90 views

Criterion for Membership in Hardy Space $H^{1}(\mathbb{R}^{n})$

Let $H^{1}(\mathbb{R}^{n})$ denote the real Hardy space (I am agnostic about the choice of characterization). It is known that if $f:\mathbb{R}^{n}\rightarrow\mathbb{C}$ is a compactly supported ...
5
votes
0answers
36 views

Almost surely, for all $s \ge 0$, there exist $t$, $u \ge s$ with $B_t < 0 < B_u$?

Let $(B_t)_{t \ge 0}$ be a Brownian motion starting from $0$. Then, do we have that, almost surely, for all $s \ge 0$, there exist $t$, $u \ge s$ with $B_t < 0 < B_u$?