Mathematical analysis. Consider a more specific tag instead: (real-analysis), (complex-analysis), (functional-analysis), (fourier-analysis), (measure-theory), (calculus-of-variations), etc. For data analysis, use (data-analysis).

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1answer
42 views

Uniformizable space implies T3 space

A topological space $(X, \mathscr{T})$ is uniformizable if there exists a uniform structure on $X$ that induces $\mathscr{T}$. I am trying to prove that every uniformizable space on $X$ is T3. To do ...
2
votes
2answers
41 views

discontinuity of an operator

I want to show that if $X=Y$ is the subspace of $L^1(0,1)$ over $\mathbb C $ consisting of all polynomials, then $T:X \times Y \rightarrow \mathbb C$ given by $T(f,g)=\int_0^1 f(t)g(t) dt$ is not ...
2
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1answer
30 views

sufficient condition for being an integral factor

Let $ f: \mathbb {R}^m \rightarrow \mathbb {R}-\{0\} $ function $C^{\infty}$ class and $w$ a one-form $C^{\infty}$ class in $\mathbb {R}^m $. If $\alpha=w-\dfrac{1}{f}dx_{m+1} $ satisfies $\alpha ...
2
votes
2answers
38 views

How to Plot $\sqrt{\frac{a^2+(b-1)^2}{a^2+(b+1)^2}}=2$

How to plot this complex division? $$ \sqrt{\frac{a^2+(b-1)^2}{a^2+(b+1)^2}}=2 $$
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4answers
125 views

Free variables in definitions

Consider the epsilon delta definition of the limit below, as it is usually stated: the limit of f as f approaches a is L if and only if, for all ε > 0, there is a δ > 0 such that for all x, 0 ...
0
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1answer
80 views

equivalent in cauchy integral for matrices

I don't know why $(zI-A)^{-1} = \frac{1}{z} \sum_{k=0}^\infty \frac{A^k}{z^k}$ in a link!
2
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4answers
36 views

Union of a set that is not compact with 0

I know that $$A=\{ \frac{1}{n}: n \,\epsilon \, \mathbb N \}\, \subseteq\, \mathbb R $$ is not compact However, I am confused why $$A\, \cup\, \{0\}$$ is compact. My attempt at understanding: Let ...
0
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1answer
46 views

Proving $x<y<z \implies \frac{f(y)-f(x)}{y-x} \le \frac{f(z)-f(y)}{z-y}$

Suppose $f$ is convex on $I$ and $(x,y,z)\in I^3$: How to prove that: $$x<y<z \implies \frac{f(y)-f(x)}{y-x} \le \frac{f(z)-f(y)}{z-y}$$
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3answers
107 views

Find $f$ if $f(f(x))=\sqrt{1-x^2}$

Find $f$ if $f(f(x))=\sqrt{1-x^2} \land [-1; 1] \subseteq Dom(f)$ $$$$Please give both real and complex functions. Can it be continuous or not (if f is real)
4
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1answer
133 views

Odd and even square roots of $z^2-1$

This is a very interesting exercise (provided that it is correct). Find two holomorphic functions $\,f_1: \Omega_1\to\mathbb C$ and $f_2:\Omega_2\to\mathbb C$, which are both square roots of $z^2-1$, ...
5
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1answer
257 views

Fourier transform of $f(x)=\frac{1}{e^x+e^{-x}+2}$

Let $$f(x)=\large \frac{1}{e^x+e^{-x}+2}$$ Compute the Fourier transform of $f$. We can factor the denominator to get $$f(x)=\frac1{(\exp(x/2)+\exp(-x/2))^2}=\frac1{(2\cosh(x/2))^2}$$ I'm thinking ...
3
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4answers
158 views

Prove that $\mu\left(\cup_{k=1}^\infty A_k\right)=\sum_{k\ge1}\mu(A_k)$

Suppose that the measurable sets $A_1,A_2,...$ are "almost disjoint" in the sense that $\mu(A_i\cap A_j) = 0$ if $i\neq j$. Prove that $$\mu\left(\cup_{k=1}^\infty A_k\right)=\sum_{k\ge1}\mu(A_k)$$ ...
2
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1answer
70 views

Show that $\sum_{n=0}^\infty a_n z^n$ converges $\forall z\in\mathbb{C}.$

Assume that $\sum_{n=0}^\infty b_n z^n$ converges $\forall z\in\mathbb{C}.$ Let $x=\lim_{ n\rightarrow\infty}|\frac{a_n}{b_n}|$ exists. Show that $\sum_{n=0}^\infty a_n z^n$ converges $\forall ...
2
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2answers
224 views

Prove that $P'$ has $n-1$ distinct real roots

Suppose a polynomial $P$ of degree $n$ has $n$ distinct real roots then $P'$ (the derivative of $P$) has $n-1$ distinct real roots. Proof by Induction: Base case: For $n=1$, $P_1 (x)=a_0+a_1x, ...
0
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1answer
103 views

Integral of holomorphic function which tends to $0$

Let $R > 0 $, $z \in \mathbb{C}, \ f : D(z,R) \rightarrow \mathbb{C} $. $Re(f) \ $ and $Im(f) \ $ are $C^{1} $ on $D(z,R) \ $. Then f is complex differentiable in $z$ if and only if $$ \lim_{r ...
0
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1answer
117 views

If $f(x)=\chi_{(0,\infty)}\exp(-1/x)$, show that $f\in C^{\infty}$.

Define the function $f:\mathbb{R}\to\mathbb{R}$ as follow: $f(x)=\chi_{(0,\infty)}\exp(-1/x)$ In other words: $f(x)=0$ if $x\le 0$, and $f(x)=\exp(-1/x)$ if $x>0$. Show that $f\in C^{\infty}$. ...
-1
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1answer
200 views

Prove that there exists a sequence of compact sets $K_1\subset K_2\subset…\subset A$ such that $\mu(A-\cup_{j\ge1}K_j)=0$.

Let $A\subset\mathbb{R}^n$ be measurable. Prove that there exists a sequence of compact sets $$K_1\subset K_2\subset...\subset A$$ such that $\mu(A-\cup_{j\ge1}K_j)=0$. Here $\mu(A)$ is the ...
3
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1answer
57 views

The statements $f(n) = O(n^{\epsilon})$ for all $\epsilon > 0$ and $f(n) = n^{o(1)}$.

Consider the statements \begin{align} \tag{A} f(n) &= O(n^{\epsilon}) \text{ for all } \epsilon > 0 \\ \tag{B} f(n) &= n^{o(1)} \end{align} Questions: It's clear that (B) implies (A). ...
1
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1answer
46 views

Existence of a nonzero vector to form

Let $ f: \mathbb {R}^m\times \mathbb {R}^m \rightarrow \mathbb {R}^m $ an alternate form of grade two. If $ m $ is odd, prove that there exists $ v\neq 0 $ such that $ f (u, v) = 0 $, for all $ u \in ...
1
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1answer
76 views

Does a barrel contain a neighborhood of $0$? [closed]

Suppose $X$ is topological vector space which is of the second category in itself. Let $K$ be a closed, convex, absorbing subset (a barrel) of $X$. Prove that $K$ contains a neighborhood of $0$.
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1answer
170 views

$\int_{0}^{+\infty}\frac{\sin x}{x^{k}(1+x^{2})}dx \ $ via residue calculus

I want to evaluate with calculus of residues $$\int_{0}^{+\infty}\frac{\sin x}{x^{k}(1+x^{2})}dx \ $$ $ k \in \mathbb{N}, k \geq 1$ If $k = 1$ we have $$\int_{0}^{+\infty}\frac{\sin ...
1
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1answer
58 views

Example of absolute continuous function that is not in $H^1$?

Consider functions on $D=[C_1,C_2]$ with real $C_1<C_2$. Could somebody please give me an example function that is absolutely continuous on D but is not in $H^1$? Many thanks!
0
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1answer
21 views

derivative with integrals proof

Suppose that $f$,$g: [a,b]→R$ are continuous and differentiable, with $f'$ and $g'$ continuous, on $(a,b)$. How do I prove that $\dfrac{d}{dx}$ $\int_{a}^x f(t)g'(t)dt$ = $\dfrac{d}{dx}[ ...
0
votes
1answer
81 views

Fundamental Domain for Congruence (mod 2) Group

How can I show that the area between the circles $|z|=1$, $|z+\frac{1}{2}|=\frac{1}{2}$, $|z-\frac{1}{2}|=\frac{1}{2}$ in the upper-half plane (here's a picture) is a fundamental domain for the ...
1
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1answer
92 views

Interior points and open sets

From what I understand, a set is open if every element or point in said set is an interior point. Now, suppose that I have a set $S$ with infinite points $s_0,s_1,s_k,...$ and so on. Mathematically, ...
0
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1answer
53 views

Pointwise Convergence with Boundness implies $L^2$ Convergence

If $f$ and $f_k$ are integrable functions to $\mathbb{R}$ on an closed interval and $\{f_k\}$ converges pointwise to $f$ with $\sup_{k\in\mathbb{N}}\Vert f_k\Vert_\infty<\infty$, I think ...
1
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1answer
56 views

proof of differentiation with integrals

Suppose that $f : [a,b] \to \mathbb{R}$ is integrable. Define, for $x \in [a,b]$, $F(x) = \int_a^x f(t)dt$ and $G(x) = \int_a^x F(t)dt$. How do I prove that $G$ is differentiable on $(a,b)$? I was ...
0
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2answers
32 views

Show that the function is continuous

Let $f:A \rightarrow R$. Suppose that for each $\epsilon>0$ there is a continuous function $g:A \rightarrow R$ so that $|f(x)-g(x)|<\epsilon$, for each $x$ in $A$. Show that $f$ is continuous. ...
0
votes
1answer
68 views

Minkowski functional and strange theorem

I have a theorem that says the following: Let X be a normed space and $U\subset X$ a convx subset with $0 \in \text{int(U)}$, then we have: $U$ is absorbing and if $\{x;||x|| < \epsilon\} \subset ...
3
votes
2answers
297 views

Show that $f, f^{-1}$ are continuous

Let $A,B \subset \mathbb{R}$ be open, and $f:A\rightarrow B$ be surjective and strictly monotonic increasing. Show that $f,f^{-1}$ are continuous. Proof: I first show $f$ is injective. Let $x,y ...
1
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0answers
100 views

Solutions to Dirichlet problem on the half space with $L^{\infty}$ boundary data.

Consider the Dirichlet problem with boundary data $f(x)\in L^{\infty}(\mathbb{R}^{d-1})$ on the halfspace $\mathbb{R}^{d}_{+}$, where $y>0$ and $x\in\mathbb{R}^{d-1}.$ One can prove that ...
3
votes
2answers
85 views

Does $L^1$ contain a subspace isomorphic to $c_0$?

Can any $L^1$ space, say $L^1(\mathbb{R})$, have some subspace isomorphic to $c_0$? I guess not but I don't see an argument right now.
2
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1answer
75 views

A question on Harmonic functions.

$f$ and $g$ are two given continuous functions on $\overline {D(0,1)}$) and both $f$, $g:\overline {D(0,1)}\to\Bbb R$ are harmonic on unit disc $D(0,1)$. Now, If $f\equiv g$ on some $C=\{e^{i\theta} ...
1
vote
1answer
118 views

Wirtinger's inequality in higher dimension

Wirtinger's inequality for one-dimensional functions states that if $f(x)$, $f'(x) = \frac{df(x)}{dx}$ $\in$ $\mathcal{L}^2(a,b)$ and either $f(a) = 0$ or $f(b) = 0$ then \begin{equation} \int_{a}^{b} ...
3
votes
1answer
96 views

Cesaro means and equivalent sequences

Let $(u_n)$ be a sequence of complex numbers that converges in mean (Cesaro convergence). Let $(v_n)$ be a sequence such that $v_n\sim u_n$. Does the sequence $(v_n)$ converge in mean? Here is ...
2
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0answers
35 views

Description of homogeneous polynomials

Let $x=(x_1, x_2,\dots ,x_d)$, prove that $span_R\{(x\cdot w)^k | w\in S^{d-1}\}$ is the set of all real homogeneous polynomials of degree k with d variables $x_1, x_2, \cdots , x_d$. ($S^{d-1}$ is ...
6
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4answers
383 views

Simply Connected domains.

If $U$ and $U'$ be two domains in $\Bbb C$, and $f$ be a homeomorphism in $U$ and $U'$ then domain $U$ is simply connected $\iff$ $U'$ is simply connected. I found this problem in complex analysis. So ...
1
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1answer
33 views

$ \int_{\{u>j\}} (u-j) dx = \int_j^\infty | \{u>j\}| dt?$

I have seen that if $u$ is a summable function (in fact, I saw that if $u \in W^{1,p}$, but I think that summable is sufficient) in $\mathbb{R}^n$ then \begin{equation} \int_{\{u>j\}} (u-j) dx = ...
11
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1answer
248 views

Integer parts of multiples of irrationals

Let $\alpha>0$ and define $$S(\alpha)=\big\{\lfloor n \alpha \rfloor: n\in\Bbb Z^+ \big\}.$$ Here $\lfloor x\rfloor$ is the integer part of $x$ and $\mathbb Z^+$ the set of positive integers. ...
0
votes
1answer
69 views

Uniformly continuous on a compact set, still uniform on a subset?

So if I have a function that is uniformly continuous on a compact set K, do all subsets of K inherit the uniform continuity? If I restrict myself to the reals, this seems to be true. But what happens ...
1
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1answer
25 views

Proving a combination of differentiability

$f:\mathbb{R}\to \mathbb{R} $where$$f(x) = \begin{cases} \dfrac{P(x)}{x^n}e^{-1/x^2}& \text{if $x\ne 0$}, \\ 0 &\text{if $x = 0$}.\end{cases}$$ Where P(x) is a polynomial and $n\geq 0$ is an ...
2
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3answers
75 views

Are the convergent sequences dense in the bounded sequences?

Since it would be comfortable for something I am currently trying to prove if this would hold I wanted to ask here whether it is true that $c$ is dense in $l^{\infty}(\mathbb{N})$?
1
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1answer
64 views

proving a function is differentiable

$g: \mathbb{R} \to \mathbb{R} $ $$g(x) = \begin{cases} x^2\sin(1/x)& \text{if $x\ne 0$}, \\ 0 &\text{if $x = 0$}.\end{cases}$$ Prove that g is differentiable everywhere, and that its ...
0
votes
1answer
57 views

sequence of series-Dirichlet Criterion

I have to check if this series : $\sum \frac{\sqrt{n+1}-\sqrt{n}}{n}$ converge.. Can I do this,using the Dirichlet Criterion?I thought that I could let $a_{n}=\frac{1}{n}$ and ...
0
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1answer
51 views

To alternatively prove the theorem(*) by proving that $g^{(n+1)}(z_0)=0$ $\forall z_0\in \Bbb C$

Assume that $g=x+iy$ be an entire function. By a theorem(*), $\vert x(z)\vert \le N \vert z\vert ^n \ \ \forall z$ large enough and for constant $N\gt 0$ and for non-negative $n\in \Bbb Z$ ...
0
votes
2answers
100 views

Cauchy integral formula in complex analysis

Assume $g$ be an entire function. And $\exists \ n\gt 0 \:and\ n\in \Bbb Z $ and also $\exists N \: and\ M \in \Bbb R$ s.t. $\forall z \in \Bbb C , \ \ \vert z\vert \ge M\ \ \: and\ \ \ \vert ...
1
vote
0answers
15 views

References for vector valued spaces like $\mathcal C^\infty(A,C_0^\infty(B))$

I'm reading a text that use the space $\mathcal C^\infty(A,C_0^\infty(B))$, where $A\subset \mathbb R^n$ and $B\subset\mathbb R^m$ are opens. I've found, in Topological Vector Spaces, Distributions ...
6
votes
2answers
121 views

Radius of convergence of a power series.

Consider the power series $\sum_{n \ge 1} a_n z^n$, where $a_n$ is the number of divisors of $n^{50}$. What is the radius of convergence? My attempt $a_n < n^{50}$ $\forall$ $n$. So $\lim ...
2
votes
0answers
366 views

Inverse of bounded self adjoint operator on HS is self adjoint?

Let $A=A^{*}$ be a bounded self adjoint operator on a Hilbert space $\mathcal{H}$ with Range Ran$(A) = D$ dense in $\mathcal{H}$. $A$ is injective, since Ran$(B) \perp ker(B^{*}) = ker(B)$. So ...
2
votes
1answer
343 views

counterexample for Dominated Convergence Theorem

The Dominated Convergence Theorem is as follows: What if the sequence $\left\{f_n \right\} \notin L^1$? Could someone provide a counterexample as to why the theorem wouldn't hold? Thanks!