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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2
votes
1answer
28 views

Show that $\lim_{n\to ∞} |a_n| = |a|$ if $a_n\to a$

Let $(a_n)$ be a convergent sequence with $$\lim_{n\to ∞} a_n = a$$. Show that $$\lim_{n\to ∞} |a_n| = |a|$$ Then state and disprove the converse statement. In order to prove that I would use the ...
5
votes
1answer
41 views

$\forall\ x,y,z\in \mathbb{R}$ Show that: $|x+y|+|y+z|+|x+z|\leq |x+y+z|+|x|+|y|+|z|$

$\forall\ x,y,z\in \mathbb{R}$ Show that: $$|x+y|+|y+z|+|x+z|\leq |x+y+z|+|x|+|y|+|z|$$ i tired, i notice that $x,y,z$ plays a symmetrical role in the inequality notice also that ...
0
votes
0answers
21 views

Contradiction proof for a limit law $f(x) \le g(x)$

Suppose that $f(x) \le g(x)$ for all $x$. Prove that $\displaystyle \lim_{x \to a} f(x) \le \lim_{x \to a} g(x)$, provided these limits exist. I posted a similar question, but this is a different ...
0
votes
0answers
6 views

The space of distribution $H^{-1}$

Let's suppose to have a function $u$ in the space $L^\infty(0,T;H^1(\mathbb{R}^n))$ with $\partial_t u\in L^\infty(0,T;H^{-1}(\mathbb{R}^n))$. So $\partial_t u$ is a linear and continuous functional ...
0
votes
1answer
15 views

Show that $F_1$ is a continuous linear functional in the normed space $(C[0,1],\|\cdot\|_\infty)$?

$(C[0,1],\|\cdot\|_\infty)$ and $$F_1(f)=\int_{\frac{1}{2}}^{\frac{4}{3}} f(t) dt$$. Show that $F_1$ is a continuous linear functional.
5
votes
1answer
39 views

Theorem 3.54 (about certain rearrangements of a conditionally convergent series) in Baby Rudin: A couple of questions about the proof

Here's the statement of Theorem 3.54 in Principles of Mathematical Analysis by Walter Rudin, 3rd edition: Let $\sum a_n$ be a series of real numbers which converges, but not absolutely. Suppose ...
0
votes
1answer
13 views

Support of a tempered distribution

Let $P(x_1,\cdots,x_n)$ be a polynomial in $\mathbb{R}^n.$ What is supp$(\widehat{P})$ when $P$ viewed as a tempered distribution. Can supp$(\widehat{P})$ be the boundary of an sphere?
0
votes
0answers
12 views

A Question about fundamental matrix of system $x'=A(t)x$

Assume in linear system $x'=A(t)x$ the coefficient matrix $A(t)$ is a periodic matrix with period $T$ and $A(-t)=-A(t)$ . If $X(t)$ be a fundamental matrix for $x'=A(t)x$ such that $X(0)=I$ then show ...
1
vote
5answers
39 views

Prove Using Definition that the Sequence $\frac{n+1}{\sqrt{n}+2}$ Diverges to Infinity

Formally, a sequence $x_n$ diverges to infinity whenever for all $M>0$ there exists $N(M)$ such that $n>N(M)$ implies $x_n>M$. Prove formally, using the definition, that the following ...
0
votes
1answer
25 views

the limit superior of a sequence exists iff the limit inferior of all subsequences of the sequence exist?

The question is nearly the same as the title, that is, the limit superior of a sequence (of real numbers) exists (can be infinity)iff the limit superior of all subsequences of the previously ...
-1
votes
1answer
72 views

Proof that it is not uniformly convergent on R [on hold]

Prove that the series $$\sum_{n=1}^\infty 2^n \sin \left( \frac{x}{3^n} \right)$$ is not uniformly convergent on R.
1
vote
2answers
25 views

Continuous increasing bounded function, derivative

Is it true that a differentiable (and hence continuous) increasing bounded function $f:\mathbb{R} \to \mathbb{R}$ has derivative $f'$ that must go to zero as $x \to \infty$. If it is, could someone ...
1
vote
3answers
42 views

About the infinite geometrical sequence factored with n [duplicate]

I just came across this thread, and i asked myself: I know that $\sum^\infty_{n=0} x^n = \frac{1}{1-x}$ But what happens when we set up the sum like $$\sum^\infty_{n=0} nx^n = ?$$ There is ...
0
votes
0answers
10 views

Proving a certain function involving the Riemann-Zeta function is non-increasing

Show that $ f(x) = \frac{\zeta(x -2)}{\zeta(x-1)} \qquad x > 3, $ where $\zeta$ is the Riemann-Zeta function, is non-increasing. My attempt was to use $\zeta(s) = \frac{1}{\Gamma(s)} ...
1
vote
1answer
35 views

Continuity of function and topology

I have this exercice $E=\{a,b,c,d\}$ with the topology $\tau=\{\emptyset, \{a\},\{a,b\},\{a,b,c\},E\},$ and the space $F=\{x,y,z,w\}$ with the topology $\theta=\{\emptyset.\{y\},\{y,z,w\},F\}$ I ...
1
vote
2answers
20 views

$f$ is continuous at $x_0=0$ if and only if $f$ is continous $\forall x\in X$?

Let $f$ be a linear functional on a normed space $(X, \|\cdot\|)$. Prove that $f$ is continuous at $x_0=0$ if and only if $f$ is continuous at every $x\in X$. I understand that the $\Leftarrow$ is ...
1
vote
0answers
16 views

Meaning of a Hypersurface resulting from Lagrange Multipliers

Suppose we have a function $f(x_1,\ldots,x_n)$ that we wish to maximize under the set of $n-1$ constrictions $g_i(x_1,\ldots,x_n) = c_i$ for $i \in \{1,\ldots,n-1\}$. We write the Lagrangian ...
1
vote
0answers
17 views

A question about fundamental matrix of periodic system $x'=A(t)x$

$X(t)$ is a fundamental matrix of linear differential equation $x'=A(t)x$ where $A(t)$ is a periodic matrix with period $T$ . Show that there exist a non-singular matrix like $C$ such that for ...
2
votes
3answers
35 views

Show that the sequence of functions $(x_n)_{n≥1}$ in $C[0, 1]$ given by $x_n(t) = t^{2n} − t^{3n} , ∀t ∈ [0, 1]$ is bounded

That is $C[0,1]$ equipped with the supremum metric. I have proven, using derivatives, that each function $x_n$ has a local maximum and local minimum at $(2/3)^{1/n}$ and $0$ respectively. I know ...
0
votes
1answer
34 views

Convergent squence in topology

Please, I consider this topological space $(E,\tau)$ where $\tau=\{G\subset E, ~\text{card}~ (E\setminus G)~\text{countable}\}\cup\{\emptyset\}$ How to prove that a sequence $(x_n)$ is convergent in ...
3
votes
0answers
44 views

Question about solutions of $x''+(1+r(t))x=0$ when $\int_1^\infty |r(t)| dx <\infty$ .

Let $x''+(1+r(t))x=0$ where $r(t)$ is continous and $\int_1^\infty |r(t)| dx <\infty$ show that the equation has solutions $\phi_1$ and $\phi_2$ such that $$\lim_{t\to\infty} ...
4
votes
0answers
20 views

Function in Lipschitz space

I'm looking for a function that is in $W^{1,1}(0,1)$ but only in the Lipschitz space $\mathrm{Lip} (\alpha, L_2(0,1))$ for $0<\alpha < 1$. $\mathrm{Lip}(\alpha, L_2(0,1))$ is defined as the set ...
0
votes
0answers
25 views

Proving that $f(z)\neq \frac{z}{z+1}$ in $D_1(0)$

Suppose $f$ is analytic in $D_r(0)$ for some $r>1$. I want to prove that $f(z)\neq \frac{z}{z+1}$ in $D_1(0)$. This is how I tried to prove this. Assume $f(z)= \frac{z}{z+1}$ in $D_1(0)$. Now ...
1
vote
0answers
32 views

Problem 30 in the Exercises following Chapter 2 in Baby Rudin: How to immitate the proof of Theorem 2.43?

Here's problem 30, the very last one, in the Exercises after Chapter 2 in Walter Rudin's Principles of Mathematical Analysis, 3rd edition: Imitate the proof of Theorem 2.43 to obtain the following ...
2
votes
2answers
58 views

Is the subset $[0, \sqrt2] ∩\mathbb{Q} ⊂ \mathbb{Q}$ closed, bounded, compact?

Letting $\mathbb{Q}$ be equipped with the Euclidean metric. What I can work out is that it is bounded as its contained in the closed ball of radius ${\sqrt2}/{2}$ centred at ${\sqrt2}/{2} $. Its not ...
0
votes
3answers
43 views

Contraction-like mapping without fixed point?

If $(X,d)$ is a complete metric space and $\xi:\;X\to X$ satisfies: $$d(x,y)<n+1\Rightarrow d(\xi(x),\xi(y))<n$$ $$d(x,y)<1/n\Rightarrow d(\xi(x),\xi(y))<1/(n+1)$$ for all $n= 1,2,\dots$, ...
3
votes
3answers
56 views

Is $\int_1^\infty \frac{\log(x-1)}{x(x-1)}\,dx$ convergent?

Does the following integral $$\int_1^\infty \frac{\log(x-1)}{x(x-1)}\,dx$$ converge? If it is convergent can we compute it?
0
votes
0answers
13 views

Two different definitions of “scale invariance”

I found the following definition of "scale invariance" from a book (http://books.google.co.kr/books/about/Critical_Phenomena_in_Natural_Sciences.html?id=rQSIZVOQfWYC&redir_esc=y) A function ...
0
votes
1answer
15 views

A simple question related to One-to-One function and linear operator

I was stuck in one line derivation about the linear operator-related question: Suppose $T$ is linear operator maps from $\mathbb{R}^n$ to $\mathbb{R}^n$. and let $c>0$ be constant. If for all ...
0
votes
1answer
17 views

Variant of Picard-Lindelof theorem

Question Let $I=[0,a]$ and define the norm $||f||_{\lambda}=\sup_I |e^{-\lambda x}f(x)|$ for $f\in C(I)$. Let $\phi:\;\mathbb{R}^2\to\mathbb{R}$ satify $|\phi(x,u)-\phi(y,v)|\leq\rho |u-v|$ for all ...
2
votes
1answer
27 views

Is it true that a quasiconvex, increasing and continous function, is convex?

Let $f:\mathbb R^n \to \mathbb R$ be a continuous and increasing function. Let $f$ be quasiconvex. Let $f(0)=0$. Can we say that $f(x)$ is convex ? If yes, how do we prove it ? Thank you very much ...
5
votes
1answer
52 views

Proof that the harmonic series is < $\infty$ for a special set..

In one of my books i found a very interesting task, i am really curios about the solution: Let $M = \{2,3,4,5,6,7,8,9,20,22,...\} \subseteq \mathbb{N}$ be a set that contains all natural numbers, ...
1
vote
1answer
31 views

Example of metric space completion

I'm looking for examples of noncomplete metric spaces and their completions. I know of some basic examples like completion of open intervals and rational numbers(both with the reals and p-adic ...
2
votes
4answers
57 views

Show the subset $A$ of $\mathbb{R}^n$ is compact

Show the subset $$A = \{(x_1, . . . , x_n) ∈ \mathbb{R}^n| −1 ≤ x_1 ≤ x_2 ≤ · · · ≤ x_n ≤ 1\} \subset \mathbb{R}^n $$ is compact, and show the function $$\left\{\begin{array}{}f : A → ...
0
votes
1answer
22 views

Uniform continuity of the function: [on hold]

Test the uniform continuity of the function $f(x)=x^{2/3}\log x$ where $x$ belongs to $(0, \infty)$.
0
votes
1answer
26 views

How can I show that if a set is bounded, then it's contained in a k-cell?

The set is a bounded subset of R (under the Euclidean metric), and a k-cell is a set of points {x_1...x_k} such that a_j < x_j < b_j for j=1...k. Any ideas on how to show this?
2
votes
1answer
34 views

Convergence of the integral of cos(x)/x^2

For $n$ in the natural numbers let $a_n = \int_{1}^{n} \frac{\cos(x)}{x^2} dx$ Prove, for $m ≥ n ≥ 1$ that $|a_m - a_n| ≤ \frac{1}{n}$ and deduce $a_n$ converges. I am totally stuck on how to even ...
1
vote
2answers
28 views

Is every Lebesgue measurable function bounded on a set of positive measure

Let $f$ be a Lebesgue measurable function from $[0,1]\to\mathbb{R}$. Let $\mu$ be Lebesgue measure. Does there exist a measurable set $B$ with $\mu(B)>0$ and an $M>0$ such that for all $x\in B$, ...
0
votes
1answer
20 views

Constructing an outer measure on a collection of subsets

Let $X$ be the set of three elements $\{a,b,c\}$ . On the collection of subsets $C = \{\{ \emptyset \} , \{a\} , \{a,b\}\}$ Define the set function $m: C → [0,∞]$ by $m( \emptyset ) = 0 $, ...
1
vote
2answers
38 views

Short proof using continuity and set conclusion

I'm new to uni math and in my most recent assignment I got stuck trying to proof the following: Let $a,b \in \mathbb{R}$ and $a<b$. Suppose $\space f:[a,b] \rightarrow \mathbb{R}$ be continuous. ...
0
votes
3answers
30 views

Find couples of complex numbers

I found this exercise, given: $$u=|z|+|u|$$ and $$z=|u|+1$$ (it is a system I don't how to write it in latex from) I have to find the couples of complex numbers $u,z$ that comes from the two equation. ...
0
votes
1answer
23 views

Finite measure and a measure $>0$?

This is a problem from one of the analysis qualifying exams in my school Let $\mu$ be a $\sigma$-finite measure on $(X,F)$ with $\mu(X) = \infty$. Show that for every $C > 0$, there exists an $E ...
0
votes
0answers
14 views

Find a matrix and a vector using partial derivative and system of matrices.

Let $f(x)$:=[$f_1(x),...,f_d(x)]^T$ and suppose that |$\frac{\partial^2 f_i(x)}{\partial x_j \partial x_k}|$$\le$K for all $i,j,k$=1,...,d and $x\in\Re^2$. Show how to define an $dxd$ matrix $J(y)$ ...
1
vote
1answer
12 views

integration concening Fourier transfom variable and space variable

We define the short time Fourier transform as follows: $$V_{g}f(x,w)=\int_{\mathbb R} f(t)g(t-x)e^{-2\pi itw} dt, (x,w \in \mathbb R).$$ (We may assume that $f$ and $g$ nice functions so that every ...
0
votes
1answer
19 views

Quadratic spline and quadratic interpolation

I am trying to understand what is the difference between quadratic spline and quadratic interpolation. Thank you for any help and advice.
0
votes
1answer
45 views

Irrational numbers in [0,1] [on hold]

Why iirational numbers in interval [0,1] can't be countable union of closed sets?
1
vote
1answer
42 views

Check if $f_n(x)=x^n-x^{2n}$ is convergent in C([0,1]) [on hold]

Check if $f_n(x)=x^n-x^{2n}$ is convergent in C([0,1]) - (continuous functions)
1
vote
0answers
21 views

Is $S(\mathbb{R}^{d})$ dense in $L^{1}_{\textrm{loc}}(\mathbb{R}^{d})$?

Let $S(\mathbb{R}^{d})$ denote the class of Schwartz functions in $\mathbb{R}^{d}$. Is it true that $S(\mathbb{R}^{d})$ is dense in $L^{1}_{\textrm{loc}}(\mathbb{R}^{d})$, the locally integrable ...
0
votes
2answers
27 views

continuity of functions on intervals

Suppose that $f : (a,b) \to \mathbb R$ is continuous. Then, there is a continuous $g : [a,b] \to \mathbb R$ such that $g(x) = f(x)$ for all $x \in (a,b)$. That is, a function defined and continuous on ...
1
vote
1answer
22 views

Continuity of $f(x)=(xI-A)^{-1}$?

Let $A\in \mathbb{C}^{n\times n}$ and $I_n$ be an identity matrix. If $z\in \mathbb{C}$ is not a eigenvalue of $A$, then $f(x)=(xI-A)^{-1}$ is a continuous function at $z$. Is that correct?