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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Proving a property of the largest limit point

Let $(a_n)_{n\in\mathbb{N}}$ be a bounded sequence. By Bolzano-Weierstraß this sequence does have a limit point. Let $\bar{a}$ denote the largest limit point of the sequence. Show that among ...
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30 views

How to see that this function is real?

I am given the function $$ \sum_{n,m \ge 0} a_n^* a_m \begin{pmatrix} n & m & 1 \\ 0 & 0 & 0 \end{pmatrix}^2, $$ where the matrix is the Wigner 3j-symbol(real quantity(!)) and is ...
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1answer
24 views

Least upper bound of an empty set

Is the following proof that the least-upper-bound of an empty set $\phi$ is $ -\infty$ correct? I'm confused because someone told me that every $x \in \mathbb{R}$ is a least-upper-bound of $\phi$. ...
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1answer
43 views

which of the following Set cannot be set of discontinuity of real valued function. [closed]

Which can not be the set of discontinuities of $f:\mathbb{R}\rightarrow \mathbb{R} $ 1) Empty set $\;\;\;\;\;\;\;\;\;$ 2) $\mathbb{Q} \;\;\;\;\;\;\;\;\;$3) Set of Irrational ...
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2answers
87 views

Help understanding proof of Theorem 2.43 in Baby Rudin

Theorem $\hspace{5 pt}$ Let $P$ be a nonempty perfect set in $\mathbb{R}^k$. Then $P$ is uncountable. Proof $\hspace{5 pt}$ Since $P$ has limit points, $P$ must be infinite. Suppose $P$ is ...
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24 views

Simple real analysis question - inf and sup

This problem looks fairly simple. I just have a question as to what is being asked specifically. The question says to find the infimum and supremum of the set A=$\{a+a^{-1}:a\in \mathbb{Q}, ...
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2answers
45 views

Definition of neighborhood

I am starting to work through Rudin's Principles of Mathematical Analysis. For $(X, d)$ a metric space and $x \in X$, Rudin defines the neighborhood $N_r(x)$ of $x$ to be the set consisting of all ...
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47 views

Classics on abstract algebra and real analysis

I am going through Apostol's calculus volume 1. What a wonderful creation from Apostol. Even I could not imagine that such a book introducing the basic concepts so informally but easy-to-understand ...
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1answer
32 views

Product of weakly differentiable functions

Let $ u,v \in W^{1,1}_\mathrm{loc}(\Omega) $ and assume that $ uv \in L^{1}_\mathrm{loc}(\Omega) $ and $ u\, Dv + v \,Du \in L^{1}_\mathrm{loc}(\Omega) $. I want to prove that $ uv \in ...
2
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2answers
50 views

Convergence of $(a_n)$ when $(a_n^{1/n})$ converges

Let $(a_n)$ be a sequence of positive numbers such that the sequence $(a_n^{1/n})$ converges. What is a sufficient condition that $(a_n)$ also converges? $a_n=n$ is an example of a divergent sequence ...
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2answers
68 views

Showing $\sqrt{0+\sqrt{0+\sqrt{0+…}}}=0$

How do I show that $a=\sqrt{0+\sqrt{0+...}}=0$ when the solution I get comes to $a^2=a \implies a \in \{0,1\} \:\: \forall a \in \mathbb{R}$? Or is it possible for $\sqrt{0+\sqrt{0+...}}$ to equal 1? ...
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2answers
55 views

Proof for distances to a set

With a metric space $(X,d)$, prove that $|d_E(x)-d_E(y)|\leq d(x,y)+d(y,z)$. In this context, $x \in X$, $d_E(x)=\inf\left\{d(x,z) : z \in E\right\}$, E is a subset of X. I've already proved the ...
2
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0answers
61 views

An entropy inequality

Let $f:[0,2\pi]\to \mathbb{R}$ be a smooth, positive function such that $f(0)=f(2\pi)$, and $\int_0^{2\pi}fd\theta=2\pi.$ Is it true that $$2\int_0^{2\pi}f\ln fd\theta- 2\int_0^{2\pi}\ln ...
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49 views

How prove there exist $(a,b)$ such $f'^2_{x}(a,b)+f'^2_{y}(a,b)-4(a^2+b^2)=0$

Question: let $D=\{(x,y):x^2+y^2<1\}$,and $f\in C^{1}(D)$,if $$|f(x,y)|\le 1 ,((x,y)\in D)$$ show that: $\exists (a,b)\in D,$$$f'^2_{x}(a,b)+f'^2_{y}(a,b)-4(a^2+b^2)=0$$ I only solve ...
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1answer
12 views

Subsequence and Accumulation Point Proof

Suppose x is an accumulation point of {$a_n : n \in J$}. Show that there is a subsequence of {${a_n : n \in J}$}$_{n=1}^\infty$ that converges to x. I understand the general idea behind the problem ...
2
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1answer
26 views

Find a discontinuous linear map on $c_0$

I want to find a discontinuous linear map $\phi: c_0 \to \mathbb{C}$. where $c_0$ has sup norm obviously, $\|.\|_\infty$ I can't think of any example. please suggest me one. I ll try checking it ...
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1answer
21 views

Completion and Seprability of C[0$\infty$]

If I have C[0,$\infty$] the space of all continuous functions on [0,$\infty$] with metric $$ \phi(\omega_1, \omega_2) = \Sigma^{\infty}_{n=1} (1/2^n)*max_{0{\leq} t {\leq} ...
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0answers
21 views

A question on $ L^1(\mathbb{R})$.

Question- Let {$f_k$} be a sequence in $ L^1(\mathbb{R})$ such that $\sum\limits_{k=1}^\infty||f_k||_1<\infty$. Prove that the series $\sum\limits_{k=1}^\infty f_k$ converges in $L^1(\mathbb{R})$ ...
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2answers
27 views

prove T is continuous and find its norm.

question- Let $x\in l^2$. Prove that $Ty=xy=(x_1y_1,x_2y_2,...)$ defines a linear map from $l^2$ to $l^1$. Also show that T is continuous and find the norm ||$T$||. How can i show ...
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1answer
38 views

Is it true that $\textrm{supp}(f)\subseteq K$ implies $f|_{\partial K}=0$?

Maybe this will be an elementary question but I need to clarify this. Let $X$ be a metric space and let $f:X\longrightarrow \mathbb R$ continuous. Suppose $\textrm{supp}(f)\subseteq K$ where $K$ is ...
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1answer
30 views

Precalculus: Analyzing Graphs: Homework

So this is the homework for my precalculus class. It’s usually fine — I end up teaching myself the relevant content because it is difficult to understand the teacher and the other students waste a lot ...
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11 views

Algorithm conjecture

This is an algorithm analysis question for computer science. Let f(n) be positive functions. (Dis)prove the following conjecture? $$f(n)+o(f(n))\in\Theta(f(n))$$ o above is the Little O, defined as ...
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1answer
45 views

Infinite Product implies divergence or not?

If $\displaystyle\prod_{n=1}^{\infty} (1-a_{n}) = 0$ then is it always true that $\displaystyle\sum_{n=1}^{\infty} a_{n} $ diverges? ($0 \leq a_{n} < 1) $
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1answer
21 views

Counterexample on weaker version of result about compact sets

The following is a very well known theorem: Let X be a metric space. $K \subset X$ is compact iff every collection $ \{ F_j \}_{j\in A}$ of closed sets with the finite intersection property in K ...
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1answer
18 views

Proof of seperability of the real/complex Lp Metric spaces

If Lp is the set of sequences of real or complex numbers, such that the infinite series, (where you sum up the moduli of the terms in the sequence to the power of p) converges to a finite value. Let ...
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1answer
30 views

Find $\textbf{x}$ such that $\left|\textbf{x}-\textbf{a}\right|=\left|\textbf{x}-\textbf{b}\right|$, for $\textbf{a},\textbf{b}\in\mathbb{R}^k$

Using the properties of inner product, it's easy to show that one solution is the midpoint, $\textbf{x}=\frac{1}{2}(\textbf{a}+\textbf{b})$. How do I show that there are other solutions, i.e., all ...
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2answers
53 views

Solve cos(z) + sin(z) = i, where z is a complex number and i the imaginary unit

So yeah everything is in the title, I tried the trigonometric identity with sin(a+bi) and cos(a+bi) and I tried changing sin(z) and cos(z) for their complex expression, but all to no avail EDIT: ...
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1answer
54 views

prove or disprove f(x+y) = f(x) + f(y) for some non-zero function [closed]

prove or disprove the following statements There is a non-zero function f: R-> R such that: f(x+y) = f(x) + f(y) for all x,y in R; f(xy) = f(x)*f(y) for all x,y in R; f(x+y) = f(x)*f(y) for all ...
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0answers
25 views

Soft Question: What are some elementary motivations of using functional analysis to study probability theory?

Recently i've become curious about the links between functional analysis and probability theory. What are some simple reasons why a functional analytic approach is preferable to a measure theoretic ...
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0answers
29 views

Approximating by smooth functions with compact support.

Consider a bounded domain $D \subset \mathbb{R}^n$ and the Sobolev space $H^1_{0}(D):=\overline{C_c^{\infty}(D)}^{W^{1, 2}(D)}$. Further, consider a Sobolev function which happens to be smooth: $u\in ...
2
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1answer
35 views

trace map is continuous

Prove that $tr: M_n(k)\to k$ is continuous. I did continuity of determinant map using induction, but how to prove trace map is continuous. please give a thorough answer. My analysis is not too good. ...
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1answer
20 views

Lim sup ratio test and Ramifications

Let $x_n$ be a real number sequence. I have managed to prove that,for a sequence $x_n$ of positive terms : if lim sup $(x_{n+1}/x_{n}) < 1$ then $x_n$ is not only eventually monotonicly ...
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2answers
17 views

approximate a Borel set by a continuous

I wonder if it is possible to approximate a Borel set by a continuous function i.e. Let $B$ a Borel set in $(X,d)$ (compact separable metric space) I wonder if there continuous functions ...
2
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1answer
54 views

How to show that a strictly decreasing, continuous function which decays slower than 1/x is not integrable?

I want to show that a function that decays slower than $1/x$ is not integrable and I tried it the following way: Assume the positive, strictly decreasing and continuous function $g(x)$ decays slower ...
4
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1answer
79 views

Why are matrices written as such?

Another thread has talked about the purpose of a matrix. Dr. Math roughly summarized it as: A matrix is just a compact notation, which allows you to specify several linear equations at once ...
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2answers
44 views

Proving the existence of fixed point $\alpha \in [-1,1]$

Can anyone help me with the following problem: I don't have the slightest idea on where to start: Consider a function $g$ which is continuous on the compact interval $[-1,1]$ such that: $g(-1)=0$, ...
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1answer
34 views

A certain relation of a polynomial to its coefficients

I've got a certain problem: If $A(t) = a_0+a_1t+ ...+a_Nt^N$, show that: $a_k = \frac{1}{2\pi}\int_{-\pi}^{\pi} e^{-ikx}A(e^{ix})dx$ after some rearrangements I got: $a_k = ...
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2answers
38 views

Name for “bicontinuous function” that's not bijective?

We know that an invertible continuous function whose inverse is also continuous is called a homeomorphism. But is there a name for a not-necessarily-bijective function that is "bicontinuous" in the ...
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0answers
28 views

Using the Lyapunov-Perron method to find the local stable/unstable manifolds

Hello Stack Exchange community. I am currently having an issue finding the local stable/unstable manifolds of this system. After going at it for a few hours I believe the person who wrote this ...
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2answers
37 views

Abel's theorem - examples

I have got Abel's Theorem in this form: If a power series is converges at one of the ends of the partition of convergence, its sum is continuous at this point (one-sided). And I have got an example ...
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22 views

dilation for general measure

I know that for the Lebesgue measure on $\mathbb{R}^n$, it holds that For $A \subseteq \mathbb{R}^n$, $$m(A + t) = m(A)$$ and $$ m(\lambda A) = \lambda^n m(A)$$ for $\lambda > 0.$ Is this ...
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25 views

Homogeneous function in Complex plane and its Periodicities

Consider A function $f:\mathbf{C}^2\rightarrow \mathbf{C}$ defined as $$f_{\alpha, \beta}(z,w)=\frac{\alpha}{z}+\frac{\beta}{w}$$ where $\alpha$ and $\beta$ both are complex number. It is easy to ...
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1answer
21 views

Sequence of irrational numbers

I have to show that: If $x$ is an real number, there is a sequence of irrational numbers converging to $x$. My attempt: We know that every $x$ real is an accumulation point of the irrational ...
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1answer
29 views

Prove that a sequence converge.

I need to do this exercise: Assume $0 \le a \le b$.Do the sequence $\{(a^{n} +b^{n})^{1/n}\}$ diverge or converge?. If the sequence converge find the limit. Well what I did is: I computed the limit ...
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0answers
29 views

Subsequences of a sequence converging and the Bolzano Weierstrass theorem

I need to prove the following: Let $\{a_{n}\}$ be a bounded sequence of real numbers.Prove that $\{a_{n}\}$ has a convergent subsequence.(Hint: You may want to use the Bolzano-Weierstrass Theorem) ...
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1answer
33 views

Prove the convergence of a sequence.

Prove that $$\left\{\frac{{n+k \choose k}}{(n+k)^k} \right\}_{n=1}^\infty \longrightarrow \frac 1{k!}$$ where $${n+k \choose k}=\frac{(n+k)!}{n!k!}.$$ My attempt for the question We only have to ...
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1answer
36 views

Prove that if a sequence $\{a_{n}\}$ converges then $\{\sqrt a_{n}\}$ converges to the square root of the limit.

My attempt and the question:Can you tell if I am right :)? thank you
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1answer
212 views

A question about a mathematical analysis book

I am a newcomer to Analysis. All knowledge I know about "Analysis" are differentials,limit and integration (basically, what we have been taught in high school) I am studying Principles of ...
4
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1answer
23 views

BMO functions are $L^p$ Loc for all $1<p<\infty$

In order to motivate my question, I'd like to remember that if $\Omega$ is a bounded domain and $f \in L^q(\Omega)$ for some $q>1$, by Hölder inequality $f \in L^p(\Omega)$ for $p \in (1,q]$ with ...
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0answers
38 views

Leibniz rule for the differentiation of an improper integral for a function with a parameter. [on hold]

I know how to prove Leibniz rule for the differentiation of a proper integral for a function with a parameter but I do not know how to completely prove the following theorem of real analysis. "If ...