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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0
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1answer
40 views

Why is $U ⊂ \mathbb{R}^n$ open with respect to metric $d_p$ iff it is open with respect to metric $d_q$ fo $q ∈ [1, ∞)$?

Let's say that for any $p ∈ [1, ∞)$ we have a distance function on $\mathbb{R}^n$ given by $$d_p(x, y) := \left(\sum^n_{j=1}|x_i - y_i|^p\right)^{\frac{1}{p}}$$ How would I show that a set $U ⊂ ...
1
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2answers
80 views

No non-trivial clopen sets in $\mathbb{R}$? How to give a direct proof?

How to give a direct proof of the following result? Let $A$ be a subset of $\mathbb{R}$ such that $A$ is both open and closed. Then $A$ is either empty or all of $\mathbb{R}$? My work: If $A$ ...
1
vote
0answers
20 views

Prob. 2, Sec. 18 in Munkres' TOPOLOGY, 2nd ed: A necessary and sufficient condition for the image of a limit point to be a limit point?

Here's Prob. 2, Sec. 18 in the book Topology by James R. Munkres, 2nd edition: Suppose that $f \colon X \to Y$ is continuous. If $x$ is a limit point of the subset $A$ of $X$, is it necessarily ...
7
votes
1answer
182 views

Evaluate $\displaystyle \sum_{n=1}^{\infty }\int_{0}^{\frac{1}{n}}\frac{\sqrt{x}}{1+x^{2}}\mathrm{d}x.$

I have some trouble in evaluating this series $$\sum_{n=1}^{\infty }\int_{0}^{\frac{1}{n}}\frac{\sqrt{x}}{1+x^{2}}\mathrm{d}x$$ I tried to calculate the integral first, but after that I found the ...
1
vote
1answer
44 views

Can't find the error in my calculation of $\int_0^\infty \frac{dx}{x^{\frac{1}{2}}(1+x^2)}$

I know $$\int_0^\infty \frac{dx}{x^{\frac{1}{2}} \left( 1+x^2 \right)} = \frac{\pi}{\sqrt{2}}.$$ I want to calculate this integral using complex analysis. So I want to integrate $$ ...
2
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3answers
50 views

Interior of the Graph of $\sin(1/x)$

Let $f(x)=\sin(1/x)$. If I'm correct then the boundary of $S=\{(x,f(x)):x\in\mathbb{R}\} \subset \mathbb{R}^2$ is $\{(x,f(x)):x \text{ in domain of } f(x)\}\cup\{(0,0)\}$, since for any $\varepsilon ...
2
votes
3answers
24 views

Proving that $\{(x,y):x^2-2x+y^2=0\}\cup \{(x,0):x\in [2,3]\}$ is connected

Let $E=\{(x,y):x^2-2x+y^2=0\}\cup \{(x,0):x\in [2,3]\}$. It appears to me from the sketch that this set is connected. However, I have no idea how to prove this, since the set is not convex, and we ...
2
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5answers
59 views

Proof for $e^{\frac{1}{n+1}}-1-\frac{1}{n}\leq0$

I'm looking for a proof of $e^{\frac{1}{n+1}}-1-\frac{1}{n}\leq0$, optionally $\ln(n)+\frac{1}{n+1}\leq \ln(n+1)$
0
votes
1answer
17 views

is the complement of first category is always second category

is the complement of first category is always second category in general space( which is not complete). I think it is true only if the space isx
1
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1answer
29 views

The proof of inverse function theorem - two questions

I have question about differentation in linear spaces. In proof of inverse function theorem I found remark about diferentation. I underlined parts wich I am interested in. source Red one I get that ...
3
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0answers
25 views

Prove convergence in measure (i.e., in probability) “distributes” over addition and respects nonnegativity.

Suppose $X_{n}$, $Y_{n}$, and $Z_{n}$ are random variables, with $Z_{n} \geq 0$ a.s. and $X_{n} \xrightarrow{p} X$, $Y_{n} \xrightarrow{p} Y$, and $Z_{n} \xrightarrow{p} Z$. Prove the following ...
1
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3answers
45 views

Showing that $\{(x, y) \in \mathbb{R}^2: x^2+4y^2\le 1\}$ is connected

Let $E :=\{(x, y) \in \mathbb{R}^2: x^2+4y^2\le 1\}$. Then there do not exist any two sets $A$ and $B$, such that $A$ and $B$ are relatively open in $E$ and $A\cup B = E$, since $E$ is closed. Hence, ...
0
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2answers
22 views

functions acting as linear functionals on their dual space

Supposing $f\in L^p$, where p and q are conjugate exponents, what does it mean that "f is completely determined by its action as a linear functional on $L^q$"? (Quoting Folland's Real Analysis here). ...
0
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2answers
31 views

Under what conditions is $\sup(A)$ not an accumulation point for $A$? $A$ is a subset of $\mathbb{R}$

I am having a hard time conceptualizing this question. I can't see any conditions that would make it soI couldn't find an accumulation point. If $m = \sup(A)$, then can't we always find some ...
0
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0answers
12 views

If A is nowhere dense in M, and if G is a nonempty open set in M, prove that A is nowhere dense in G.

If A is nowhere dense in M, and if G is a nonempty open set in M, prove that A is nowhere dense in G. I tried by contradiction but could not figure it out. I found that we can use following result ...
0
votes
1answer
36 views

How to find the roots of a equation involving log terms?

This question was in my test and I am not sure what to do with it let $f:(0,\infty)\rightarrow \mathbb{R}$ be given by $$f(x)=\log x-x+2$$ then its number of roots of $f$are. So putting it ...
1
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3answers
45 views

Recommend book Taylor expansions

I've taken up self-study of math and i start using the book called : Mathematical Analysis I Authors: Canuto, Claudio, Tabacco, Anita I would like to start from zero to understand taylor ...
0
votes
3answers
101 views

Very hard integral limit

$$ \lim_{n \to \infty }\int_{0}^{\pi} x^n\sin x \mathrm{d}x$$ I have stumbled across this problem in an old book and havent managed to figure out how to solve it by using basic and intermediate ...
0
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0answers
14 views

Sequence of sequence Best notation (Functional Analysis)

In functional analysis, one often deals with "sequences of sequences", what is the best notation for them? For example in $l^2$, each element $x=(a_1,a_2,\dots)$ is a sequence. So if we talk about a ...
0
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1answer
29 views

Prove that a point x not belonging to a closed set M \subset (x,d) always has a nonzero distance from M

I am sure this proof is pretty straight forward and I know that I somehow will need to show that $x$ is in $A^\prime$ if and only if $D(x,A) = 0$, where $A$ is any nonempty subset of $X$. Thank you ...
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0answers
23 views

Holomorphic function $f$ such that $f(z) = -icz + d$ for real constant $c$ and complex constant $d$

I'm trying to solve the following problem: Suppose that $f = u + iv$ is holomorphic on the set $A =$ { $z \space | \space Re(z) > 1$ } and that $\partial u/ \partial x + \partial v/ \partial y = ...
0
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0answers
28 views

$\lim\limits_{n\to\infty}$ of iterated function $e^{x-1}$

Do you think there's a better way to do this? Given that $E(x)=e^{x-1}$, $n$ iterates of $E(x)$ are defined as $E(\underbrace{E(E(...(x))...)}_{n-1\text{ times}}=E^{\circ n}(x)$. Here's my attempt ...
0
votes
3answers
31 views

$e^{x-1}$ has only one fixed point

How does one show that the function $E(x)=e^{x-1}$ has only one fixed point? We know that there is only one integer solution for $e^{1-x} = x$, which is $x=1$, but Wolfram Alpha also gives a second ...
0
votes
1answer
28 views

How can I find the norm

Let $v $ be a $2\times 2$ matrix as follow: $$v= \begin{pmatrix} \frac{\partial^2 u}{\partial x^2} & \frac{\partial^2 u}{\partial x \partial y}\\ \frac{\partial^2 u}{\partial x \partial y} ...
0
votes
1answer
27 views

Uniform Continuity of a Complex Function

Let $f:D\to\mathbb{C}$ be a complex function $$f(z)=\exp\left(-\frac{1}{z^2}\right)$$ with $D=\{z:0<|z|\le R\}$ for some $R$. The results from numerical experiment suggest that this function is not ...
1
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2answers
53 views

Let $\mu_n$ be a sequence of finite measures on space $(X,M)$ and $\forall E \in M, \lim_{n \to \infty }\mu_n(E)=\mu(E)< \infty $..

Let $\mu_n$ be a sequence of finite measures on space $(X,M),M-\text{ sigma algebra on X}$ and $\forall E \in M, \lim_{n \to \infty }\mu_n(E)=\mu(E)< \infty $ and let $f$ be a bounded function. ...
0
votes
1answer
28 views

Is the set satisfying $-1 \leq Re(z) \leq 5$ closed?

I want to prove if such set is closed. We define for $z = x+iy$ in $\mathbb{C},$ $$\mathcal{O} := \{z \in \mathbb{C}: -1 \leq Re(z) \leq 5\} = \{(x,y) : x \in [-1,5], y \in (-\infty,\infty)\}.$$ ...
1
vote
1answer
38 views

What's the derivative of $f(x)=(\frac{x^2 + 1}{x^2 + 3})^{\sin(2x)}$

I'm trying to calculate the derivative of $f(x)=(\frac{x^2 + 1}{x^2 + 3})^{\sin(2x)}$ using the difference quotient but somehow I don't succeed. Help and elegant solution appreciated!
2
votes
1answer
53 views

Given that $f_n \to f$ in $L^1(\Omega)$, $\mu(\Omega )=1$ and $ \|f_n\|_2^2 \leq M$, show $ \|f\|_2^2 \leq M$.

Given that $$\int_{\Omega} |f_n -f | \, d \mu \to 0,$$ $\mu(\Omega )=1$ and $ \|f_n\|_{L^2}^2 \leq M$, show $ \|f\|_{L^2}^2 \leq M$. Attempt: Note first that $f_n \in L^1(\Omega)$ since $$\|f_n ...
0
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2answers
44 views

Conversion of $\cos(x)$ to $\frac{\exp(ix)+\exp(-ix)}{2}$

just out of fun I started playing around with the sums to derive $\cos(x)$ from $\frac{\exp(ix)+\exp(-ix)}{2}$ $\frac{\exp(ix)+\exp(-ix)}{2}=\frac{1}{2}\sum_{k=0}^\infty \frac{(ix)^k+(-ix)^k}{k!}$ ...
4
votes
1answer
44 views

$\int_{\Omega} |f_n-f||f_n| \, d \mu \to 0$ if $f_n \in L^1(\Omega)$, $f_n \to f$.

Suppose $f_n \to f$ in $L^1(\Omega)$ where $\mu(\Omega)=1$. Suppose $$\int_{\Omega} |f_n| \, d\mu \leq M$$ for all $n$. Is there a way to show that the integral $$\int_{\Omega} |f_n-f||f_n| \, d ...
1
vote
1answer
27 views

Let $(X, M, \mu )$ be a space with measure. $f:X \to \mathbb R \text{ and } f\in L^1(X).$ ..

Let $(X, M, \mu )$ be a space with measure. $f:X \to \mathbb R \text{ and } f\in L^1(X).$ Prove that for all $\epsilon > 0$ that there exists $\delta > 0$ such that for $E \in M$, $\mu(E)< ...
1
vote
2answers
85 views

Proving $x_{n+1}=\frac{1}{2}\left( x_n+ \frac{2}{x_n}\right)$ is a Cauchy Sequence

I want to prove that the sequence given by: $$x_0=1$$ $$x_{n+1}=\frac{1}{2}\left( x_n+ \frac{2}{x_n}\right)$$ is a Cauchy sequence. In order to do this, my approach was to prove that it is a ...
0
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0answers
19 views

If $E \subset [0,1]$ satisfies, for any $I \subset [0,1]$, $m(E \cap I) \geq \frac{1}{2}m(I)$, then $m(E)=1$

If $E \subset [0,1]$ satisfies, for any $I \subset [0,1]$, $m(E \cap I) \geq \frac{1}{2}m(I)$, then $m(E)=1.$ I'm aware this post exists elsewhere, say, here but what I don't understand is why we ...
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0answers
28 views

Why is causality important for laplace transformations? [closed]

Could someone please explain why causality is important for laplace transformations?
1
vote
0answers
25 views

Showing an inequality of LimSup

Let $\{x_n\}_{n=1}^\infty$ and $\{y_n\}_{n=1}^\infty$ be sequences of real numbers. Verify that each of the following holds, provided the right-hand side makes sense. ...
0
votes
1answer
37 views

Proving that $f(x)=\frac{1}{x^2 \ln x} $ is Lebesgue measurable on $(2, + \infty)$

I have that a set $E$ is Lebesgue measurable if the outer measure: $$\mu^*(E)=\inf_{I_1,...,I_n} \mu (I), E \subseteq I_1 \cup I_2 ,...\cup I_n , I_i-\text{intervals}$$ satisfy the three properties ...
6
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0answers
157 views

Brezis Exercise 3.27 extension.

Let $E$ be a separable Banach space with norm $\|\cdot\|$. The dual norm on $E^*$ is also denoted by $\|\cdot\|$. Let $(a_n) \subset B_E$ be a dense subset of $B_E$ with respect to the strong ...
24
votes
4answers
1k views

Were “real numbers” used before things like Dedekind cuts, Cauchy sequences, etc. appeared?

Just the question in the title, I'm trying to understand how something like analysis could be developed without formal constructions of the real numbers. I'm also very interested, if the answer is ...
0
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2answers
43 views

If $\sum_n x_n$ converge to $L$, what is $\lim_{n \to \infty} x_n$? [duplicate]

If $\sum_n x_n$ converge to $L < \infty$, what is $\lim_{n \to \infty} x_n$? This is a question I have been asked, but I cannot answer it. Can someone explain it to me?
10
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0answers
178 views

Finding an example of nonhomeomorphic closed connected sets

Question: Find two closed, connected subsets in $\mathbb{R}^2$, $A$ and $B$, such that $A$ is not homeomorphic to $B$, but there is a continuous bijection $f:A \rightarrow B$ and a continuous ...
0
votes
2answers
29 views

Show that there exist $\{x_n\}$ a sequence of elements in $A- \{x\}$ such that $x_n \to x$

Let $E$ a normed vector space and $A \subset E$. Let $x$ an accumulation point in $A$. Show that there exist $\{x_n\}$ a sequence of elements in $A- \{x\}$ such that $x_n \to x$. Definition : An ...
0
votes
1answer
27 views

A proof in Hilbert & Courant vol 1 of Weierstrass theorem.

My question is regarding a derivation of an inequality on page 67 of Methods of Mathematical Physics. Here's a scan of the book: ...
0
votes
3answers
31 views

Generated sigma algebra and its countable subcollection [duplicate]

Let $\scr{C}$ be a collection of subsets. Prove that if $A \in \sigma(\scr{C})$ (sigma algebra generated by $\scr{C}$), then there exists a countable subcollection $\scr{C}_A$ of $\scr{C}$ such that ...
0
votes
2answers
22 views

Why is $A=\{\frac{m}{2^n} \mid m, n\in \mathbb{N}, \quad 1\leq m\leq 2^n-1\}$ dense in [0, 1]?

I am stuck on this problem Why is $A=\{\frac{m}{2^n} \mid m, n\in \mathbb{N}, \quad 1\leq m\leq 2^n-1\}$ dense in [0, 1]? thanks for any help.
2
votes
5answers
60 views

Finding $\sum_{k=1}^{\infty}k^2 \frac{2^{k-1}}{3^k}$

Finding $$\sum_{k=1}^{\infty}k^2 \frac{2^{k-1}}{3^k}$$ I think if $$\int_{1}^{\infty}x^2 \frac{2^{x-1}}{3^x}dx$$ exists that this sequence is convergent, but I doubt that this integral is equal to ...
1
vote
2answers
48 views

Prob. 19, Chap. 1 in Baby Rudin: For what $\mathbf{c}$ and $r > 0$ does this equivalence hold?

Here's Prob. 19 in Chap. 1 in the book Principles of Mathematical Analysis by Walter Rudin, 3rd edition: Suppose $\mathbf{a} \in \mathbb{R}^k$, $\mathbf{b} \in \mathbb{R}^k$. Find $\mathbf{c} \in ...
0
votes
0answers
12 views

constructions over rotation surfaces in $\mathbb{R}^3$

Let $f: (0, \infty) \times \mathbb{R} \to \mathbb{R}$ be continuously differentiable, and $\nabla f(x) ≠ 0$ on the set $M = f^{-1}(0)$. (Which means that $M$ is a 1-dimensional manifold of the ...
1
vote
2answers
65 views

Continuity of Popcorn Function (Thomae's Function)

I have to prove that the function $f:]0,1] \rightarrow \Bbb R$ : $$ f(x) = \begin{cases} \frac1q, & \text{if $x \in \Bbb Q$ with $ x=\frac{p}q$ for $p,q \in \Bbb N$ coprime} \\ 0, & \text{if ...
1
vote
0answers
32 views

Prob. 16, Chap. 1 in Baby Rudin

Here is Prob. 16, Chap. 1 in the book Principles of Mathematical Analysis by Walter Rudin, 3rd edition: Suppose $k \geq 3$, $\mathbf{x}, \mathbf{y} \in \mathbb{R}^k$, $\vert \mathbf{x} - ...