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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2answers
69 views

Construct a harmonic function that appears to be discontinuous on the unit circle.

Construct a harmonic function $u$ in $D(0,1)$ that satisfies $$ lim_{r \to 1^-}u(re^{i\theta}) = \begin{cases} 1 & 0 < \theta < \pi \\ 0 & \pi < \theta < 2\pi \...
2
votes
0answers
56 views

Are there any “default” properties which hold for almost all topological spaces in analysis?

What is the simplest/most general commonly used (type of) topological space in analysis? For instance, every example I can think of in analysis is first-countable. I don't think it can be metric ...
0
votes
0answers
12 views

Let $Q=[0,1]\times[0,2]$. Find the rotation matrix $(A)$for the angle $\frac{\pi}{4}$ upon this set. Is $A(Q)$ measurable and a elementary set?

Let $Q=[0,1]\times[0,2]$. Find the rotation matrix $(A)$for the angle $\frac{\pi}{4}$ upon this set. Is $A(Q)$ measurable and a elementary set? First off, I know that $A$ is a linear map, and a ...
4
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1answer
30 views

$\mathcal{l}^1$ is not complete for the norm $\|\cdot\|_\infty$

Let $\mathcal{l}^\infty = \{ (u_n) | u_n \in \mathbb{R}$ and $sup_{n \in \mathbb{N}}|u_n| < \infty \}$ and $\mathcal{l}^1 = \{ (u_n) | u_n \in \mathbb{R}$ and $\sum_{n=1}^{\infty} |u_n| < \infty ...
1
vote
1answer
12 views

Functions composition commutativity

I have to prove that $\circ$ is not, in general, a commutative operation of Funct(X,X). My approach: Let X be a set, $a,b\in X$, $a\neq b$ constants. Let $i,j \in Funct(X,X)$ with $i:X \to X,\text{ } ...
0
votes
1answer
25 views

Prove or disprove: $\lVert T\rVert=\sup_{\lVert x\rVert=1}|\langle Tx,x\rangle|$, where $H$ is a Hilbert space and $T$ is bdd linear operator.

Edit: To clarify, note that $T:H\to H$. This is a problem on an old preliminary exam in Analysis that I'm working through to prep for my own prelim. My initial thought was to disprove it, but I can'...
0
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1answer
31 views

Good, simple reference for Riesz-Fischer Theorem.

I am looking for a good, simple reference for the proof of Riesz-Fischer Theorem ($L^p$ spaces are complete). An example of a not so good reference in my opinion is Royden, where he uses "rapidly ...
2
votes
2answers
53 views

Preimage of a function

I'm having difficulties with the notion of preimage, specifically with this example: Let $A$ be a subset of $[0, 1]$. We define $$f(x) = \begin{cases} x, & x \in A; \\ -x, & x \in [0, 1] \...
0
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3answers
47 views

Finding limit of this function

Given function: f(x) = $\sqrt{x^2+x}-x$ I used 3th binomial formula and brought it to this form: $\frac{x}{\sqrt{x^2+x}+x}$ But now no idea how to get limit of this (goes to ∞). By testing I know ...
0
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3answers
184 views

Evaluate the integral $ \int_0^{+\infty} \frac{\sin(x^2)}{x^4+1} dx $ using the residue method

I have a problem in evaluating the integral above. So far I've proceeded in this way. We have an even function, so: $$ \int_0^{+\infty} \frac{\sin(x^2)}{x^4+1} dx = \frac{1}{2} \int_{-\infty}^{+\...
0
votes
1answer
30 views

Infinite union of countable sets proof.

I understand how to prove that the union of 2 countable sets is countable. I then began to think we can use induction to say that the countable union of countable sets are also countable. However my ...
0
votes
1answer
61 views

Why is $E$ measurable

I have some queries about this proof (towards the last part of this text). 1) Firstly, why is $E=\{x\in X:g(x)<\infty\}$ measurable? 2) Secondly, why is $\mu(X\setminus E)=0$? Is it because $\|g\|...
0
votes
0answers
29 views

Square root of several variables analytic function

The question is as follow: Let $H\subset \mathbb{C}^n$ be an simply-connected region. If $f$ is a nowhere vanishing analytic function on $H$, with $f(z)>0$ for all $z\in H\cap\mathbb{R}^n$, ...
1
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0answers
22 views

Show that change of variable theorem holds for $\cup A_i$

Problem assumes the following theorem $\mathtt{Theorem}$ Let A be an open subset of $\mathbb R^n$ and $\phi:A \to \mathbb R^n$ a one-to-one continuously differentiable map whose Jacobian $J\phi$ ...
-2
votes
3answers
55 views

Limit of type 0/0 without L'Hopital

I am trying to figure out the limit as $h \rightarrow 0$ of $$\frac{x\sqrt{x+h+1}-x\sqrt{x+1}}{h(x+h)}$$ without using l'Hopital's rule. I have tried to extend the fraction by the conjugate of ...
0
votes
1answer
26 views

$f'_n(x)$ is bounded and $f_n(x) \to 0$ for each x, then $f_n(x) \to 0$ uniformly

I want to show the question If $f_n(x)$ is differentiable on [a.b] with $|f'_n(x)|<10$ for all n and if $f_n(x) \to 0$ at each x, then $f_n(x) \to 0$ uniformly. I think I should use triangle ...
3
votes
4answers
53 views

$\lim\limits_{n\to \infty}\sum_{k=1}^{\infty}2^{-k}\sin(k/n)=0$

I want to show that $$\lim\limits_{n\to \infty}\sum_{k=1}^{\infty}2^{-k}\sin(k/n)=0$$ I first thought if I can change the order of limit, it can be easy to show that. But I found that there ...
0
votes
0answers
16 views

Finding a branch of the complex logarithmic function $\log(1-z).$

I have a question that asks me to find the holomorphic branch $L(1 − z)$ of $\log(1 − z)$ valid in the cut-plane $z \in \mathbb{C}\setminus [1, ∞)$ and such that $L(1) = 0.$ We have defined the ...
0
votes
0answers
19 views

Convergance of the average of a convergant complex sequence

So this is in exercise 3.14 (Neat!) in Baby Rudin, which I have found quite a simple and obvious proof for, however when I checked the answers the proofs I found were quite complicated so now I am a ...
1
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0answers
35 views

Almost found the limit points of this set

I want to find all limit points of the given set, and I think I almost got it. $M=\left\{\frac{x}{2^y} \mid x, y\in ℕ , x \leq y\right\}$ also $M \subset ℝ$. We say that: $\forall \delta > 0,\...
1
vote
2answers
43 views

Partial fraction decomposition of $\pi\cdot \tan(\pi z)$

Evaluate the partial fraction decomposition of $\pi \tan(\pi z)$ $$2\pi \tan(\pi z)=\cot\left(\frac{\pi}{2}-\pi z\right)-\cot\left(\frac{\pi}{2}+\pi z\right)$$ $$=\frac{2}{1-2z}+\sum_{k=1}^\infty \...
0
votes
0answers
24 views

How do I describe the analytic completion to the algebraic closure of $\mathbb{F}_2$?

Is it possible? I'm using the algebra generated by the set $\{0, 1^r\}$ for all fractions $r$ as my representation of the algebraic closure to $\mathbb{F}_2$. I can't seem to find a metric on it. ...
0
votes
0answers
33 views

Does $\min\{x_1x_3+x_2^2 | x_1^2+x_2^2+x_3^4 = 4\}$ has an optimal solution?

Does $\min\{x_1x_3+x_2^2 | x_1^2+x_2^2+x_3^4 = 4\}$ has an optimal solution? I think continuous function over closed and bounded domain has an optimal solution but I am not sure. Can anyone give me ...
0
votes
2answers
43 views

Finding all limit points of a set

How can I find all the limit points of this set? $S=\left\{\frac{x}{2^y} \mid x, y\in ℕ , x \leq y\right\}$ with $S \subset ℝ$. Could this be proved if I showed that $∀δ > 0, \exists z ∈ S\text{...
0
votes
1answer
42 views

Urysohn's extension theorem

Currently I am working my way through Ernest Michael's first article on continuous selections. Here, Urysohn's extension theorem is stated as follows: For a $T_1$-space, the following properties ...
1
vote
1answer
30 views

How to find all limit points of this set?

How to find all limit points of this set? $S=\left\{\frac{x}{y} \mid x, y\in ℕ , x \leq y\right\}$ also $S \subset ℝ$. Is the way to proof this done same way as it is for sequences? I have never ...
-8
votes
0answers
154 views

Show that map is norm preserving and determine Eigenvalues [duplicate]

Can someone of you give me a solution for this? Let $N\in \mathbb N$. a) We define the map $\mathfrak F:(\mathbb C^N, ||\cdot||_2)\to(\mathbb C^N, ||\cdot||_2)$ by $$(\mathfrak F(x))_k := \frac{...
0
votes
2answers
36 views

Is $f_n=x^n$ weakly convergent in $(\mathscr C[0,1],\lVert\cdot\rVert_\infty)$?

This is part of an old preliminary exam in Analysis I am working through. For earlier parts of the problem I have already shown that $f_n$ does not converge in $(\mathscr C[0,1],\lVert\cdot\rVert_\...
0
votes
0answers
15 views

Alexandrov Maximum Principle and $W^{2}_p$ estimates

I'm reading an article of N. V. Krylov: About an example of N. N. Ural'tseva and weak uniqueness for elliptic operators, Nonlinear partial differential equations and related topics, 131–144. This ...
1
vote
1answer
28 views

Prove that exists a linear continuous functional satisfying…

Let $E$ be a normed space over the field of real numbers. I have to prove that given two convex sets $A$, $B$ in $E$, with positive distance between then, there exists a linear continuous functional ...
2
votes
1answer
43 views

Convergent + divergent $\to$ divergent

Given sequences $(x_n)$, convergent, but $(y_n)$ is divergent, then $(x_n + y_n)$ is divergent. I am confident that it is true, but having trouble getting the formalities correct. I have tried proof ...
2
votes
1answer
53 views

Is it possible to construct such a function in analytical form?

Suppose $f\left(f\left(x\right)\right)=\sin(x)$ Is it possible to find $f$ in closed form, or any other forms so as to visualize $f(x)$ on $x\in[-\pi,\pi]$? Is it possible to prove the existence and ...
0
votes
1answer
31 views

Does an unbounded gradient implies a non-vanishing hessian determinant

Let $\Phi:\mathbb{R^{n-1}\to R}$ be a vector function. Suppose that exists a set $S_1\subset \mathbb R^{n-1}$ on which $G=\nabla\Phi$, the gradient of the function is unbounded. Does that imply ...
1
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4answers
57 views

The continuity of function's restrictions implies the continuity of function.

Let be $X \subset F_1 \cup F_2$, where $F_1$ and $F_2$ are closed. If the function $f\colon X \longrightarrow \mathbb{R}$ is such that $f|_{X \cap F_1}$ and $f|_{X \cap F_2}$ are continuous, so prove ...
0
votes
0answers
48 views

Convergence of $\sum a_n b_n$

In Rudin P.M.A The partial sums $A_n$ of $\sum a_n$ form a bounded sequence; i.e. $b_0\ge b_1\ge b_2\ge\cdots\ge b_n$ so that $\lim\limits_{n\rightarrow\infty}b_n=0$. Then $\sum a_n b_n$ converges. ...
0
votes
2answers
54 views

Equivalence relations and commutative diagrams

Let $\sim$ and $\dot\sim$ be equivanlence relations on the sets X and Y respectively. Suppose $f \in Y^X$ is such that $x \sim y$ implies $f(x) \dot\sim f(y)$ for all $x,y \in X$. Prove that there is ...
0
votes
1answer
69 views

Show: $(X,d_X)$ is complete $\Leftrightarrow $ $f(X)$ is closed in $(Y, d_Y)$ ($f: X \to Y$ is an isometric embedding)

I have the following task: Show that a metric Space $(X,d_X)$ is complete if and only if for every isometric embedding $f: X \to Y$ in another metric Space $(Y, d_Y)$ it holds true that $f(X)$ is ...
0
votes
1answer
19 views

Bounding an exponential integral

I'm having trouble seeing this bound I've seen on a proof. Let $f$ be a polynomial, and $F$ the polynomial obtained from $f$ by replacing each coefficient by its absolute value. Then: $$\bigg{|}\...
0
votes
1answer
19 views

What is $(A_1 \times … \times A_n) \cup(B_1 \times … \times B_n)=?$ ,$A_i$'s are intervals

What is $(A_1 \times ... \times A_n) \cup(B_1 \times ... \times B_n)=?$ ,$A_i$'s are intervals $[a_{Ai},b_{Ai}]$ and $B_i$'s are $[a_{Bi},b_{Bi}]$ respectively. What I mean is can $(A_1 \times ... \...
1
vote
0answers
24 views

Prob. 4, Sec. 4.3 in Kreyszig's functional fnalysis text: Application of the Hahn Banach Theorem

Let $X$ be a real or complex vector space, and let $p \colon X \to \mathbb{R}$ be a real-valued function satisfying $$p(x+y) \leq p(x) + p(y) \ \mbox{ for all } \ x, y \in X$$ and $$p(\alpha x) = \...
1
vote
1answer
50 views

Difference of functions inequality

In one book on complex analysis I see the following: But $f$ is continuous at the point $z$. Hence, for each positive number $\varepsilon$, a positive number $\delta$ exists such that $$\lvert f(...
0
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0answers
39 views

Proof about Dedekind Theorem

Following two pics are proof regarding Dedekind Theorem My question is: after prove there is a rational number r greater than $\beta$, we arrived at contradiction, why?
0
votes
1answer
40 views

If $\langle f(x),g(x)\rangle = 0$, and $f$ has periodic orbit, then $g(x)$ has equilibrium point

Let $x'=f(x), x'=g(x)$ be two ODE, with $f(x),g(x):\mathbb R^2\rightarrow\mathbb R^2$, such that $\langle f(x),g(x)\rangle =0$ for all $x\in\mathbb R^2$. If $f$ has a periodic orbit then $g$ has ...
0
votes
3answers
88 views

Isometric isomorphism between $R^2$ and $R$

Can someone help me solving the following problems? $(\mathbb R^2,d_2)$ and $(\mathbb R, d_1)$, $d_2, d_1$ being the respective euclidean norms, are not isometric isomorphic, i.e. there is no ...
0
votes
0answers
27 views

A problem involving Poincaré-Bendixson Theorem

This is the problem: Given the ODE $x'=f(x)$, suppose that $f:\mathbb R^2\rightarrow\mathbb R^2$, and $f$ satisfies the hipoteses of Poincaré-Bendixson Theorem, if exists $V:\mathbb R^2\rightarrow\...
0
votes
0answers
56 views

Spivak problem 26-4 (Calculus 3rd edition)

I am having difficulty understanding a problem requirements from Spivak's for Chapter : "Complex Functions". The problem descriptions is as follows: In this problem we will consider only ...
0
votes
0answers
22 views

How to prove a continuous function is uniformly continuous on a compact set using BW theorem?

Question. Let S be a compact set in $\mathbb R^n$ and $f:S\rightarrow \mathbb R^m$ be a continuous function. Prove that $f$ is uniformly continuous on S. I want to prove it using Bolzano-Weierstrass ...
2
votes
1answer
53 views

Functional Inequality given $s\cdot f(t)+t\cdot f(s)\leq 2$

Question: For all continuous $f:\mathbb R\to \mathbb R\ $ and $\forall s,t \in [0,1]$ that satisfies: $$s\cdot f(t)+t\cdot f(s) \le 2$$ a) Prove that: $$\int_0^1 f(x) \,dx \le {\pi/2}.$$ b) How many ...
0
votes
1answer
19 views

$d_1, d_2$ are metrics on $X$; $(X,d_1)$ is complete. Let $i:(X,d_1)\to(X,d_2)$ be continuous and $i^{-1}$ unif. cont. Show $(X,d_2)$ is complete.

This is a problem on an old preliminary exam in Analysis I'm working through. The problem initially looked easy to me; my plan is to show that for any $\{x_n\}$ Cauchy in $(X,d_2)$, we have that $\{i^{...
-4
votes
1answer
49 views

Hausdorff metric, Ultra metric

Can anyone prove the following statement though it seems simple. Let $(X,d)$ be an ultrametric space and $A$ and $B$ are closed, bounded subsets of $X$. Then for each $a$ in $A$ and $\varepsilon > ...