Mathematical Analysis generally, including Real Analysis, Harmonic Analysis, Complex Variable Theory, the Calculus of Variations, Measure Theory, and Non-Standard Analysis.

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39 views

Counterexample to Peano's theorem in infinite dimension

Would you like a counter example that Peano's theorem does not apply to spaces with infinite dimension. Peano theorem: Let E be a space with finite dimension, consider a point $(t_0,x_0) \in \Re ...
1
vote
1answer
36 views

The Gaussian Integral

Hi I am trying to calculate the expected value of $$ \mathbb{E}\big[x_i x_j...x_N\big]=\int_{-\infty}^\infty x_ix_jx_k...x_N \exp\bigg({-\sum_{i,j=1}^N\frac{1}{2}x^\top_i A_{ij}x_j}-\sum_{i=1}^Nh_i ...
2
votes
1answer
39 views

Show that the follow function is Riemann integrable on $[0 , 2]$, and use te definition to find $\int_0^2f.$

Show that the follow function is Riemann integrable on $[0 , 2]$, and use te definition to find $\int_0^2f.$ $$ f(x) = \left\{ \begin{array}{c} -1, &0 \le x < 1 \\ 2, &1 \le x \le 2 ...
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0answers
33 views

Pointwise convergence and uniform

There is a norm $\Vert .\Vert$ in the space $C([a,b],\mathbb{R})$ such that convergence pointwise implies convergence in norm $\Vert .\Vert$ ? I think not because if there would be the natural ...
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0answers
16 views

Describe the Unit Ball

I was asked to describe the unit ball in $C(I)$. All I could come up with was that by definition $B_{1}(0):=\{x \in C(I) : ||x||_{\infty}<1 \}$, where $||x||_{\infty}:=\sup_{t \in I} |x(t)|$. Thus, ...
1
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1answer
12 views

Finding all continuous solutions to an integral

I need help with both parts of this problem. Part (i) seems obvious, because the integrand $f(t)$ would become $F(t)$, which is obviously differentiable because its derivative is $f(t)$ by ...
7
votes
2answers
79 views

Constructing reals: Prove $i$ not real

So I need to prove, from the definition of reals as Cauchy sequences of rationals, that $i$ is not a real number. The guidance given is to assume that $a\sim b$ are equivalent Cauchy sequences of ...
1
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1answer
25 views

An upper bound for a strictly increasing function

Let us start with a definition: A function $\alpha \colon \mathbb{R}_{\geq 0} \to \mathbb{R}_{\geq 0}$ is of class-$\mathcal{K}_\infty$ ($\alpha \in \mathcal{K}_\infty$) if it is continuous, zero at ...
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0answers
8 views

How to prove the set of fourier multipliers is a banach algebra?

Hi I am new here at math stack Exchange, this is my first question, hope you guys can help me out:) Suppose $F\colon L^2(\mathbb{R} ) \to L^2(\mathbb{R})$ is the Fourier transform given by ...
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1answer
27 views

Understanding why $\int_\gamma {dz \over z - a} = k 2\pi i$ for $\gamma$ a closed curve not passing through $a$

The following is a paraphrased proof from Ahlfors. I bolded the part that is confusing me and asked a question about it at the bottom of this post. Hypothesis: Let $\gamma$ be a closed curve that ...
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1answer
35 views

Is there a more concise way?

I am currently trying to prove Taylor's Theorem, this is the proof I am given: I am finding this very long and hard to follow, can anyone give me a more concise, clear proof? Thanks
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1answer
10 views

What is the integrand of $\int_\gamma d\ \log(z-a)$?

Suppose $\gamma$ is a piecewise differentiable closed curve that does not pass through the point $a \in \mathbb{C}$. I'm reading a proof in Ahlfors that shows under this condition we will obtain $$ ...
11
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2answers
87 views

Second derivative of $f(f(\cdots f(x)\cdots )?$

For convenience, let's write $f_n(x)=f(f(\cdots f(x)\cdots )$ where $f$ is iterated $n$ times. Suppose: $$f(0)=0,\quad f'(0)=\alpha,\quad f''(0)=\beta$$ What is $f''_n(0)?$ I've found ...
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1answer
18 views

consequence of Mean Value Theorem

Let $f$ a continuous function on $[a, b]$ $a < b$ ,derivable on $(a, b)$ then there exist $c_1, c_2 \in (a, b)$ ,$c_1 \ne c_2$ such that $\frac{f (b) - f (a)}{b - a} = \frac{f '(c1) + f' ...
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2answers
52 views

The set $S=\{(x,y) \in \mathbb{R}^{n} \times \mathbb{R}^n = \mathbb{R}^{2n} ; x \neq y\}$ is connected if $n \geq 2$.

When n = 1 it is easy to see that is not connected, it just take the split open $ S=U_1 \cup U_2$ such that $U_1 = \{(x,y) \in \mathbb{R}^2 ; x > y\}$ is $U_2 = \{(x,y) \in \mathbb{R}^2 ; x < ...
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1answer
21 views

Prove that $f$ is differentiable on $\Bbb R$ and find the derivative.

$$f(x) = g(x)|g(x)|$$. I know that to prove that a function is differentiable, I need to prove that $$\lim_{x \to c} \frac {f(x) - f(c)}{x-c}.$$ And then to prove that the function is ...
-1
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1answer
38 views

Advanced Calc proof help

Assume that for $a,b>0$ and any $0 < t< 1$ $$ a^tb^{1-t} ≤ ta+(1-t)b $$ Prove given $a_1,a_2,...,a_n ≥ 0$, $b_1,b_2,...,b_n \geq 0$ and $b_1+b_2+...+b_n=1$ We have $$ \left(\sum_{i = ...
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vote
1answer
39 views

Proving the chain rule by first principles

I'm currently trying to prove: $(f(g))'(a)=f'(g(a))*g'(a)$ I have been given a proof which manipulates: $f(a+h)=f(a)+f'(a)h+O(h)$ where $O(h)$ is the error function. However, I would like to have a ...
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4answers
82 views

Why does the result follow?

How does this theorem follow? Theorem. If $g$ is differentiable at $a$ and $g(a) \neq 0$, then $\phi = 1/g$ is also differentiable at $a$, and $$\phi'(a) = (1/g)'(a) = -\frac{g'(a)}{[g(a)]^2}.$$ ...
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1answer
20 views

Prove this monotone sequence has a bound, thus it converges.

Let $r>0$ and $\frac{r^n}{n!}$ Prove that it converges. I know that it is eventually decreasing, so it is monotone. How do I get a bound for it to show that it converges? Also how would I go ...
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0answers
28 views

Proving that limit

Given $V=H_0^1(\Omega)$ with $\Omega$ is a bounded open domain in the plane $\mathbb{R}^2=\{x=(x_1,x_2):x_i\in \mathbb{R}\}$ and $u,v,\delta \in V$. A functional $K:\mathbb{R}\to \mathbb{R}$ such that ...
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2answers
23 views

a Fourier transform (sinc)

let $K(u) = \frac{\sin(u)}{\pi u}$ show that Fourier transform of $K$ is $ \hat{K}(\omega) = \textbf{1}_{|\omega|\leq 1} $ Some help would be appreciated
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2answers
25 views

Analytic $F(z)$ has $f(z)$ as derivative $\implies$ $\int_\gamma f(z)\ dz = 0$ for $\gamma$ a closed curve

Hypothesis: Suppose that $F(z)$ has $f(z)$ as a derivative. Suppose further that $F(z)$ is analytic. Now consider the complex line integral $$ \tag{1} \int_\gamma f(z)\ dz $$ Question: Does this ...
2
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1answer
24 views

Intregation by parts

Let $u=(-R_2\theta,R_1\theta)$ where $R_1,R_2$ are the usual Riesz Transforms in $\mathbb{R}^2$, $\mathbb{T}^2$ denotes the torus, $\theta\in C^{\infty}(\mathbb{T}^2)$ and ...
0
votes
2answers
29 views

Continuity of 1/x

I am confused with what $8(ii)$ wants from me, I answered the first part of this question with help from the question posted here Is $f(x)=1/x$ continuous on $(0,\infty)$? But the this proves ...
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1answer
22 views

Proving Uniform Continuity using Bolzano Weierstrass

I have been working on this question for some times, and can't seem to put together the contradiction needed using Bolzano. any help would be greatly appreciated,
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0answers
10 views

Hilbert-Schmidt theorem

In the Hilbert-Schmidt theorem what it means : $A e_n=\lambda_n e_n$ ? Thank you .
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2answers
73 views

What is the sufficient and necessary condition for changing the order of summation?

What is the necessary and sufficient condition for $\sum\limits_{i=0}^{\infty }{\sum\limits_{j=0}^{\infty }{{{a}_{ij}}}}=\sum\limits_{j=0}^{\infty }{\sum\limits_{i=0}^{\infty }{{{a}_{ij}}}}$? Suppose ...
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vote
1answer
37 views

Need helping proving that something is differentiable but not continuously differentiable

I need some help please proving that a function is differentiable at $(0,0)$ but not continuously differentiable at $(0,0)$. The function is as follows... (from $\mathbb{R}^2$ to $\mathbb{R}$) ...
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1answer
30 views

Prove that this sequence converges

I need to show that $ |r^n|$ converges for $0<|r|<1$. I know this converges to $0$. The problem that I have is that I need to use the observation that $\lim_{x\to inf}|r^{n+1}|=\lim_{n\to ...
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1answer
39 views

Separability of functions with compact support

Let $X$ be a locally compact metric space which is also $\sigma$-compact. Let $C_{c}(X)$ be the continuous functions on $f$ from $X$ to $\mathbb{R}$ with compact support. Is $C_{c}(X)$ separable? My ...
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2answers
27 views

Determining why $\int_{\partial R} z\ dz = 0$ and $\int_{\partial R} z\ dz = 0$ independently of Cauchy's Theorem for a Rectangle

Let $R$ be a rectangle on the complex plain and $\partial R$ its closed curve. Without making use of Cauchy's Theorem for a Rectangle (or any of the other Cauchy theorems), I'm curious why we know ...
0
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1answer
23 views

Property of Projection Operator

Let $p: \mathbb{R}^n \rightarrow X$ be the projection mapping onto $X \subset \mathbb{R}^n$ compact convex. I am wondering if $$ \left( p(x) - p(y) \right)^\top z \leq \left( x -y\right)^\top z $$ ...
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0answers
23 views

What is the value of those limits?

$$\lim_{n\longrightarrow{\infty}}{\int_{0}^{\infty}{\arctan{(nx)}e^{-x^n}dx}}$$ And $$\lim_{n\longrightarrow{\infty}}{\int_{0}^{+\infty}{(1+\frac{e^{-nx}}{\sqrt{x}})(1-\tanh{(x^n)})dx}}$$
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votes
3answers
50 views

Let $T : \mathbb{R}^3 \to \mathbb{R}^3$ be a linear trasformation. Find $T(x)$

Let $T : \mathbb{R}^3 \to \mathbb{R}^3$ be a linear trasformation with $T \left(\begin{bmatrix} 1 \\ -2 \\ -1 \\ \end{bmatrix}\right) = \begin{bmatrix} 1 \\ -1 \\ 2 \\ ...
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votes
1answer
13 views

Uniform Convergence of Series Help

Suupose the sequence $(b_k) , k\geq 0$ satisties $\sum k|b_k| < \infty$, then show that $\sum_{k=0}^\infty b_kx^k$ converges uniformly to a function $g$ on $|x| \leq 1$ and that $g'(x) = ...
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2answers
27 views

Show that the difference quotient of $1/x^n$ exists

Let $n>0$ be a positive integer. For all $x\not=0$, prove that $f(x) = 1/x^n$ is differentiable at $x$ with $f^\prime(x) = -n/x^{n+1}$ by showing that the limit of the difference quotient ...
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0answers
54 views

Real Analysis and dynamics

I am looking for a textbook or similar resource that addresses the content in a rigorous graduate course in real analysis(at the level of Rudin/Royden) with the following criteria: No hand waving - ...
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0answers
10 views

Computing ${\partial U \over \partial x}$ and ${\partial U \over \partial y}$ for $U(z)= \int_\gamma (z - a)^n\ dz$

Goal: Let $$ U(z)= \int_\gamma (z - a)^n\ dz $$ I'm trying to compute ${\partial U \over \partial x}$ and ${\partial U \over \partial y}$. Attempt: I know that $(z-a)^n$ is the derivative of ...
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1answer
20 views

How to know that $(z-z_0^3)(z-z_0^5)(z-z_0^7) = \sum_{k=0}^3 z^k z^{3-k}_0$

How to know that $$(z-z_0^3)(z-z_0^5)(z-z_0^7) = \sum_{k=0}^3 z^k z^{3-k}_0$$ with $z_0$ a root of $z^4+1$. I can check that it is true, but is there a way to tell, by seeing the LHS expression, ...
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2answers
29 views

Infinite series of a function involving an enumeration of rationals on $0,1$

Let $\{ q_n : n \in \mathbb{N} \}$ be an enumeration of the rational numbers in $(0,1)$ and define $f_n(x) = \begin{cases} 0 \qquad \text{if} \; x \in (0, q_n) \\ 2^{-n} \quad \: \text{if} \; x \in ...
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1answer
40 views

Deducing Laplace Formulas

I have to compute the followings integrals $\forall\; b\in \mathbb{C},\; \text{Re} \;b \gt0,p\gt 0$ $$ \int_{-\infty}^\infty \frac{e^{ipx}}{x-ib}$$ $$ \int_{-\infty}^\infty \frac{e^{ipx}}{x+ib}$$ ...
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1answer
21 views

Definition of an integrand

General Question: Say we have integral $$ \int f(z)\ dz $$ Is the integrand in this context (i) $f(z)$ or (ii) $f(z)\ dz$? In any case, is $f(z)\ dz$ a formally defined mathematical object in its ...
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2answers
43 views

Derivative of a Matrix to a Power

Fix a positive interger $k$ and let $F: \mathbb{R}^{n \times n} \rightarrow \mathbb{R}^{n \times n}$ be the map on $n \times n$ matrices defined by $F(A)= A^k$. Show that $F$ is differentiable at ...
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1answer
32 views

some problem about countable I think? [on hold]

I cannot solve this: Let $\left \{ G_{\alpha } \right \}_{\alpha \in A}$ be a collection of open sets in $\mathbb{R}$ such that $\bigcup _{\alpha \in A}G_{\alpha }=\mathbb{R}$. Show that there is a ...
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3answers
27 views

Some problem about Cauchy sequence.

I cannot solve this: Let X be a complete metric space with a metric d. (a) Suppose that the sequence $x_{n}$ in X satisfies $\sum_{n=0}^{\infty}d(x_{n},x_{n+1})<\infty$ Show that $x_{n}$ ...
0
votes
1answer
25 views

What are the values of the parameters that make the function differentiable at zero?

I think I might have found a way to solve this problem but I'm not sure if this is correct, if someone could tell me if this is the correct approach or not that would be nice. If it's not the correct ...
0
votes
0answers
20 views

about lp estimate of schwarz function

A homework question that I really couldnt find how to start. Prove that for any f in schwarz class $ \lVert f \rVert_{q} \leq C_{p,q} \lVert \nabla f \rVert_{2}^{a} \lVert f \rVert_{2}^{1-a} $ $ ...
1
vote
1answer
39 views

Prove by using step functions: $\int_{-b}^{b}\sin(x)\ dx = 0$

The Assignment: Let $b > 0$. Prove by using step functions: $$\int_{-b}^{b}\sin(x)\ dx = 0$$ The claim itself is obvious, but I have no idea how to prove it with step functions. My idea was ...
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vote
1answer
20 views

Is the inverse of a symmetric positive semidefinite matrix also a symmetric positive semidefinite matrix?

If we let $$S_{++}^n(\mathbb{R})$$ denote the set of all square symmetric positive definite matrix over the real numbers, then is it true if $A\in S_{++}(\mathbb{R}) \implies A^{-1} \in ...