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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Sequence of increasing compact sets

Suppose $X$ is a locally compact metric space which is $\sigma$-compact. Let $K$ be a compact subset of $X$. We can find a sequence of compact sets $K_{n}$ such that $K_{n} \subset \textrm{int}(K_{n + ...
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Series $\sum_{n=0}^\infty (-1)^n \frac{x^{4n+1}}{4n+1}$

Does anyone know the sums of the following two series? $$\sum_{n=0}^\infty (-1)^n \frac{x^{4n+1}}{4n+1}$$ $$\sum_{n=1}^\infty (-1)^{n+1} \frac{x^{4n-1}}{4n-1}$$ I encounter such series in my work.
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Inverse of the function $\frac{(1+x)^2-i(1-x)^2}{(1+x)^2+i(1-x)^2}$

It can be proved that the function $f:[-1,1]\to \mathbb{C}$ defined by $$f(x)=\frac{(1+x)^2-i(1-x)^2}{(1+x)^2+i(1-x)^2}$$ maps the interval $[-1,1]$ one to one onto the lower part of the unit circle. ...
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this function is in some Holder space?

Consider $\Omega$ a open, bounded, and smooth domain of $R^N$ with $N \geq 3.$ And let $f: \Omega \times R \rightarrow$ a Caratheodory function . Supoose that $f$ is locally Lipschitz. Supose that ...
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measurability restriction operator

Let $M\subset \mathbb{R}^k$ compact. For every $x\in M$ we define $L(x): \mathbb{R}^m \rightarrow \mathbb{R}^m $ a linear isomorphism Let $G_n (\mathbb{R}^m)=\{ W: W\ \mbox{is subspace of} \ ...
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430 views

compact set always contains its supremum and infimum

Let $K$ be a compact subset of $\mathbb R$. Prove that $\sup K$ and $\inf K$ exist and are in $K$. My approach: As $K$ is compact, it is bounded. So $\sup K$ and $\inf K$ exists. The reason is that: ...
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Deriving that ${d \over dz}\left(\log\ z \right) = {1 \over z}$ in the complex plane

How does one derive that $$ {d \over dz}\left(\log\ z \right) = {1 \over z}\text{?} $$
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33 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 ...
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1answer
33 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 ...
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76 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 ...
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33 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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24 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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1answer
16 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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1answer
8 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 ...
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15 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, ...
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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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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
9 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 $$ ...
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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
32 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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35 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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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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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 ...
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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 ...
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1answer
37 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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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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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
35 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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1answer
17 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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22 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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Proving that the terms of the sequence $(nx-\lfloor nx \rfloor)$ is dense in $[0,1]$.

I have been doing a basic math course on Real analysis...I encountered with a problem which follows as " Prove that $na \pmod1$ is dense in $(0,1)$..where $a$ is an Irrational number , $n\ge1$... I ...
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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 ...
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Does there exist a real Hilbert space with countably infinite dimension as a vector space over $\mathbb{R}$?

Essentially what the title says - where to me a Hilbert space is a complete (Hermitian) inner product space, am I safe to assume every such real Hilbert space is of uncountable dimension over ...
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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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63 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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1answer
21 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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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
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Dual Mapping Preserves Linear Independence if and only if Original Mapping is Surjective

Here is my question: Let $V$ and $W$ be finite-dimensional vectors spaces over a field $F$ and $f:V \rightarrow W$ a linear map. Show that $f$ is surjective if and only if the image under ...
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$\Delta u = f, f \in L^q \Rightarrow u \in W^{2,q}$ References

I'm looking for references for the following theorem. I will very grateful Theorem: [Calderón Zigmund] If $u$ is a solution of \begin{equation} \Delta u = f \quad \mbox{in} \quad B_2 ...
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Show that $\liminf\limits_{n\rightarrow\infty}a_{n}/a_{n-k}\leq \liminf\limits_{n\rightarrow\infty}a_{n}^{k/n}$

Here is an exercise: Let $\{a_{n}\}$ be a positive increasing sequence, can we prove that: $\liminf_{n\rightarrow\infty}\frac{a_{n}}{a_{n-k}}\leq \liminf_{n\rightarrow\infty}a_{n}^{k/n}$? Could ...
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Discontinuous for rationals

Show that $f\left(x\right):=\sum_{n=1}^{\infty}\frac{\left\{nx\right\}}{n^2}$, where $\left\{nx\right\}$ is the fractional part of $nx$, is discontinuous for all rationals. I guess it would be nice ...
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1answer
36 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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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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1answer
26 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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178 views

What is the *standard duality argument?

What is the standard duality argument? I saw this foor exemplo in the following statement. The case $p < 2$ follows from the standard duality argument. To prove Theorem: [Calderón Zigmund] If ...
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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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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 ...
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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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3answers
48 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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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 ...