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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Power Series Problem.

Suppose $\sum a_nx^n$ has a finite radius of convergence, say $R$, and $a_n\ge 0$ for all $n$, show that if the series converges at $R$ then it also converges at $-R$ . What I did: Applying the root ...
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4 views

Non periodic Fourier Series Point Convergence

If $f$ is a real-valued non-periodic continuous function that is differentiable at the point $x_0$, is it true that $S_n(f(x_0))$ converges to $f(x_0)$, where $S_n$ denotes the partial sums of the ...
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1answer
15 views

Prob. 4, Sec. 3.4 in Erwine Kreyszig's INTRODUCTORY FUNCTIONAL ANALYSIS WITH APPLICATIONS

Let $(e_n)$ be an orthonormal sequence in an inner product space $X$. Then, for every $x \in X$, we have $$ \sum_{n=1}^\infty \vert \langle x, e_n \rangle \vert^2 \ \leq \ \Vert x \Vert^2.$$ Now ...
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14 views

Prob. 3, Sec. 3.4 in Erwine Kreyszig's INTRODUCTORY FUNCTIONAL ANALYSIS WITH APPLICATIONS: How to derive the Schwarz inequality?

Let $\left( e_n \right)$ be an orthonormal sequence in an inner product space $X$. Then for every $x \in X$, we have $$ \sum_{n=1}^\infty \left\vert \langle x, e_n \rangle \right\vert^2 \ \leq \ ...
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10 views

Limits of Taylor POlynomials over $k$-tuples?

Let $f \in \mathscr{C}^{(m)}(E),$ where $E$ is an open subset of $R^{n}$. Fix $\textbf{a}$ $\in E$, and suppose $\textbf{x}$ $\in R^{n}$ is so close to $\textbf{0}$ that the points \begin{equation*} ...
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0answers
9 views

weak solution to one dimension conservation law

Suppose $u:\Bbb{R}\times[0,\infty)\to\Bbb{R}$ is a continuous function such that for all $v\in C_c^\infty(\Bbb{R}\times[0,\infty))$ $$ \int^{\infty}_0 \int^{+\infty}_{-\infty} \Big(u(x,t) ...
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12 views

An example in which the Fubini theorem is inapplicable

This is example 8.9(a) in Rudin's Real and Complex Analysis, (alternatively, exercise 10.2 in Rudin's Principles of Mathematical Analysis). Let $X$ and $Y$ be the closed unit interval $[0,1]$, let ...
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2answers
50 views

The norm $\|f_n-f\|_{L^1} \to 0$ but $f_n \not\to f$

A classmate and I are studying this following question from Stein-Shakarchi, Chapter 2, Exercise 12: Show that there are $f \in L^1(\mathbb{R}^d)$ and a sequence $\{f_n\}$ with $f_n \in ...
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0answers
14 views

Prove integral belongs to dual space and find operator norm [on hold]

Prove that $L: C([0,1])\to \mathbb{R}$ defined by $$L(f)=\int_{0}^{1}f(x)dx$$ belongs to $C([0,1])^{\star}$. What is its operator norm? ($C([0,1])^{\star}$ is the dual space/conjugate space).
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Proving uniform approximation by polynomials when sets are not compact

Here are two problems of the same flavor (and hence I posted them simultaneously) based on the Stone-Weierstrass Approximation Theorem. Let $f$ be continuous on $[1,\infty)$ with ...
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2answers
58 views

Proving that $ f: [a,b] \to \Bbb{R} $ is Riemann-integrable using an $ \epsilon $-$ \delta $ definition.

Problem. Show that a bounded function $ f: [a,b] \to \Bbb{R} $ is Riemann-integrable if and only if for every $ \epsilon > 0 $, there exists a $ \delta > 0 $ such that for any partition $ ...
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1answer
19 views

Area of a surface of revolution about the y-axis-

I'm trying to find the area of a surface of revolution generated by the curves $$y=x^3,\quad x=1,\quad x=2, \quad\rm{around} \quad y=-1 $$ \begin{array}{lcl} A &=& 2\pi \int_1^2 {(y + 1)\sqrt ...
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16 views

For which $(x_1,x_2)$ is this a solution to the minimal surface equation?

Let $u(x_1,x_2):=arcosh(\sqrt{x_1^2+x^2})$ then I want to find out for which $(x_1,x_2)$ this is a solution to the minimal surface equation in two dimensions that you can find for example here. ...
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1answer
22 views

Show space C([0,1]) with norm integral is a Banach space [duplicate]

Is the space C([0,1]) with the norm integral from 0 to 1 of |f(t)|dt a Banach space?
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29 views

Connected Sets Examples

(a) Give an example of a connected set $A \subset \Bbb R^n$ such that $\Bbb R^n\setminus A$ is not connected. (b) Give an example of a compact set $K \subset \Bbb R^n$ which is not connected. So far ...
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2answers
65 views

Equality of a quadratic function

Let $f: \mathbb{R}\rightarrow \mathbb{R}$ an arbitrary function and $g: \mathbb{R}\rightarrow \mathbb{R} $ a quadratic function with the following property: For any $m$ and $n$ the equation ...
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1answer
70 views

To prove $\sin(x) > x - \frac{x^3}{6}$ is strictly increasing [on hold]

How do I prove this is strictly increasing? $ \sin(x) > x - \frac{x^3}{6} $ where $ x>0 $ What I did so far, I first rearrange the inequality, Let $ f(x)=\sin(x) - x + \frac{x^3}{6} ...
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0answers
20 views

Creating a sequence convergent to zero with special characteristic

Let $\{a_k\}$ and $\{b_k\}$ be positive sequences in $\mathbb{R}$ that both converge to zero. Can we choose $\{c_k\}$ such that it converges to zero and $$ 0<\lim_{k \to \infty} \frac{a_k}{c_k} = ...
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1answer
23 views

Problems on Divergence theorem

I am struggling in the following problem: $ S \subset R^3$ is a region in divergence theorem. $\vec{n}$ is outward normal to the surface of $S$. Then, what does $div \vec{F}=0$ mean in the ...
2
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1answer
31 views

If E is measurable, then $\delta E$ is measurable.

Problem: If $\delta =(\delta_1,\delta_2,\cdots,\delta_d)$ is a d-tuple of positive numbers $\delta_i>0$, and $E$ is a subset of $\mathbf{R^d}$, we define $\delta E$ by $\delta E = ...
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1answer
48 views

How to prove that $|a_{1}+2a_{2}+…+na_{n}| \leq 1.$

Let $$f(x)=\sum_{k=1} ^{n}a_{k}\sin(kx)$$ where $n \in \mathbb{Z^+}$ and $a_{k} \in \mathbb{R}$ for each $k=1,…,n.$ Suppose that $ \vert f(x)\vert \leq \vert \sin(x) \vert $ for every $x.$ Prove ...
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1answer
21 views

Integration of step functions

I've managed parts (a) and (b) fairly easily, but c is causing me a real headache. I've seen the Cauchy-Schwartz inequality before, but I've hit a roadblock because I've no idea whether or not I can ...
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1answer
28 views

What is $\sum_{i,k=1}^n x_k^2x_i^2$?

I stumbled over the double sum $$\sum_{i,k=1}^n x_k^2x_i^2$$ and was wondering whether this is anything that can be expressed in terms of the euclidean norm? Maybe it is even $||x||^4$ but I am not ...
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1answer
25 views

Area of surface [duplicate]

What is the area of the region that is bounded by the curve $$\vec{R}(t)=(\cos^3t, \sin^3t), 0\leq t<2\pi?$$ I have no idea how to start here or what i have to use.
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45 views

Riesz's 1909 proof of the Riesz Representation Theorem

Frigyes Riesz originally proved the Riesz Representation Theorem on $ C[0,1] $ -- here is his 1909 paper in English (original French). He builds a real valued function $ \text{A} $ on $ [0,1] $ ...
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1answer
24 views

Convolution of an integrable function an $L^\infty$ function [duplicate]

Let $f$ be an integrable function on $\mathbb{R}$, and $g$ be an $L^\infty$ function on $\mathbb{R}$. Then, the convolution $f*g$ is said to be continuous and bounded on R. I managed to show that it ...
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1answer
26 views

Suppose $A$ is a nonempty subset of $\mathbb{R}^n$. Prove that if $A$ is both open and closed, then A=Rn.

I know that we want to prove if $A$ does not equal the empty set then $A= \mathbb{R}^n$. By assuming $A$ can't be the empty set we choose an $a$ in $A$ such that $B(a,r) \subseteq A$ . But $A$ is ...
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4answers
71 views

Prove that if $A$ is both open and closed, $A=\mathbb R$. [duplicate]

Suppose $A$ is a non-empty subset of $\mathbb R$. Prove that if $A$ is both open and closed, $A=\mathbb R$. I think I'm supposed to assume that $A$ is not equal to $\mathbb R$ and derive a ...
2
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2answers
30 views

Convergence of improper integral $\int_{0}^{1}\frac{\log(x)}{1-x^2}dx$

Find whether the integral converges or diverges. $$\int_{0}^{1}\frac{\log(x)}{1-x^2}dx$$ I simplified it to $$\int_{0}^{1}\frac{\log(x)}{(1-x)(1+x)}dx$$ Here I have $2$ "bad" bounds (both $0$ and ...
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2answers
29 views

Finding the radius of convergence for the given series.

How to find out the radius of convergence for $\sum x^{n!}$? I tried the ratio and the root test. But no luck. Kindly help !
2
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1answer
31 views

Area form a differential form?

I just read that $\omega_x( \eta, \zeta) := \langle x, \eta \times \zeta \rangle $ is the area form on the sphere, where $x \in \mathbb{S}^2$ and $\eta,\zeta \in T_xM.$ All I see is that this is ...
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0answers
20 views

Proving Weierstrass Approximation Theorem on just bounded sets in $\mathbb R$

Suppose $f$ is a continuous function on $\mathbb R$. Show that we can approximate $f$ uniformly by a sequence of polynomials on any bounded subset of $\mathbb R$. My attempt is as follows: ...
2
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1answer
11 views

Function of mean square continuous process

I have been asked to prove that, if $\{X_t\}$ is a ($n$-dimensional) mean square continuous process and $f:\mathbb{R}^n \rightarrow \mathbb{R}^d$ is a Lipschitz function, the process $\{f(X_t)\}$ is ...
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2answers
35 views

a discontinuous function the square of which is continuous

give an example of a discontinuous function the square of which is continuous. The domain is $[0,1]$. I tried to use the indicator function of rationals, but its square is not continuous. EDIT:I am ...
3
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1answer
22 views

Heat kernel properties

I'm having problem with the heat equation in $\mathbb{R}^n$; specifically in proving the following: let $f\in L^1(\mathbb{R}^n)$ and: $$ u(x,t)=H_{\sqrt{t}}\star f(x)=(4\pi ...
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1answer
44 views

Prove $f'''(x) \geq 3$ for some $x \in (-2,2)$, if $f$ is cont on $[-2,2]$ and three times differentiable in $(-2,2)$ & $f(2)=-f(-2)=4$ & $f'(0)=0$

Prove that there exists a $x \in (-2,2)$ such that $f'''(x) \geq 3$, if $f$ is cont on $[-2,2]$ and three times differtiable in $(-2,2)$ with values $f(2)=-f(-2)=4$ & $f'(0)=0$. How do I handle ...
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3answers
28 views

Find some n such that $|s-s_n|< 10^{-3}$

Consider the series $\sum_{n=1}^\infty \frac{1}{n^2}$. Let $s_n$ be the $n$th of the series and $s$ be the sum of the series. Find some $n$ such that$$|s-s_n|< 10^{-3}$$ Can someone please ...
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1answer
22 views

Riesz Projection as a Cauchy type integral

Let \begin{equation*} f(\zeta)=\sum_{k\in\mathbb{Z}}\widehat{f}(k)\zeta^k \end{equation*} be a complex-valued function on unit circle $\mathbb{T}=\{ \zeta\in\mathbb{C}:|\zeta|=1\},$ where ...
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1answer
30 views

What is the difference between uniform convergence and dominate convergence theorem?

I saw that both have aim to change limit with integral... that's the part that interests me most. I saw in some cases where we couldn't use uniform convergence, we use dominate convergence theorem to ...
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1answer
38 views

Validity of Implicit Function Theorem

In essence, IFT says: If f(x,y) is C' and f(a,b)=0, then f(x,y)=0 is an identity (f(g(y),y)=0). But proof depends on f(x,y)=0 in a neighborhood of b. How do you know this? If you assume f(x,y)=0, ...
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1answer
41 views

$\lambda f(x)+(1-\lambda)f(y)=f(\lambda x+(1-\lambda)y)$ implies $f $ linear?

Let $X$ be a Banach space. $f:X \to X$ a continuous function. If we assume that $f$ satisfies the following convexity condition: $$\lambda f(x)+(1-\lambda)f(y)=f(\lambda x+(1-\lambda)y),$$ for all ...
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4answers
31 views

Find $\sup_{x\in[0,1]} \frac{x}{x^2+n^2+1}$

We have $f_n:[0,1]\to \mathbb{R},\:f_n(x)=\frac{x}{x^2+n^2+1}$ and we need to prove that is uniform convergence using formula: $\lim _{n\to \infty } \sup_{x\in[0,1]} |f_n(x)-f(x)| =0$ First ...
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0answers
27 views

Why $\int_{\mathbb R^n}e^{-\sum_{i=1}^n x_i^2} \, dx_1\cdots dx_n=\omega_{n-1}\int_0^\infty e^{-r^2}r^{n-1} \, dr$

Let $\omega_{n-1}$ the area of the surface of the sphere in $\mathbb R^n$, i.e. the surface of the set $\mathcal S=\left\{(x_1,\ldots,x_n)\mid \sum_{i=1}^n x_i^2=1\right\}.$ We know that the area ...
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2answers
58 views

Convergence of $\sum_{n=1}^\infty (2n^{10}+4n^5+1)/(4n^{15}+4n^{12}+5)$

Test whether the following series converges: $$\sum_{n=1}^{\infty}\frac{2\cdot n^{10}+4\cdot n^5+1}{4\cdot n^{15}+4\cdot n^{12}+5}$$ $1.$ I know that it does not make sense to use the root ...
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0answers
36 views

Conditional expectation over a convex set

Let $\boldsymbol{X}$ be an $\mathbb{R}^d$-valued absolutely continuous and integrable random vector. Further, let the cdf $F$ of $\boldsymbol{X}$ be strictly increasing in each component on ...
1
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1answer
39 views

Characterization of normed space

Let $\mathbb R^n$ be vector space over $\mathbb R$. Then we know that all norms over $\mathbb R^n$ are homeomorphism. Is it true for $\mathbb Q^2$ over $\mathbb Q$ For instance are the Euclidean norm ...
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0answers
13 views

“Transference” Argument

In the proof of the Iwaniec-Martin theorem (giving a bound in $L^p$ for the Riesz transform, $\|R_j\|_p=\cot(\frac{\pi}{2p^*})$ the proof of this equality is given by proving the inequalities $\leq$ ...
2
votes
1answer
27 views

Example 5, Sec. 23 in Munkres' TOPOLOGY, 2nd edition: What is the closure of this set?

What is the closure in $\mathbb{R}^2$ of the set $$ \left\{ \ x \times y \ \in \mathbb{R} \times \mathbb{R} \ \colon \ x > 0, \ y = \frac{1}{x} \ \right\}? $$ I know that each point of the set is ...
0
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0answers
13 views

Commutation of Convolution, Restriction and Differentiation

Let $B$ be the open unit ball in $\mathbb R^n$ centered at zero and let $K=\bar{B}\cap (\mathbb R^{n-1}\times\{0\})$. Suppose you are given $u\in C^{1,\alpha}(B)$ such that $u|_K=f\in C^2(K)$. For a ...
2
votes
1answer
47 views

Evaluate if $f_{_n}$ converge uniformly or not

We have $f_n:[1,2]\to \mathbb{R},\:f_n(x)=\frac{x^n}{x^n+1}$ and we have to see if the convergence is uniform or not. From what I understand we need to prove that $\lim _{n\to \infty } ...