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
32 views

Uniform convergence on $[0,1]$

Let $f_n(x)=x^n$. The sequence $\{f_n(x)\}$ converge pointwise but no uniformly on $[0,1]$. Let $g$ be continuous on $[0,1]$ with $g(1)=0$. Prove that the sequence $\{g(x)x^n\}$ converge uniformly on ...
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22 views

a question about functional analysis, and this question is about Riesz's theorem(how to prove it?)

Riesz's theorem(Actually I am not quite sure whether it is Riesz theorem or not): Let $(V,\|\cdot\|)$ be a normed vector space, and suppose $C$ is a subset of $V$, moreover, $C$'s interior is not ...
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0answers
14 views

Elementary proof of Hölder´s inequality (by convexity)

I am trying to prove the Hölder´s inequality but following these steps: 1) Let $f:(a,b)\to\mathbb R$ be double differentiable then $f$ is convex if and only if $\;\;f''(x)>0$ for all $x\in (a,b)$ ...
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1answer
19 views

Homeomorphism between lower limit topology and another topology

Given a basis $B$ for a topology $T$ on R with $B=\left\{[a,b): a,b\in R-\left\{0\right\} \cup \left\{(-x,x): x>0\right\}\right\}$. Show that $(R,T)$ is homemorphic to the lower limit topology ...
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29 views

A simple non-linear equation.

Let ${\rm a}$ and ${\rm b}$ be two given vectors in ${\mathbb R}^n$. Find ${\rm u}\in {\mathbb R}^n$ and $x\in[0,\infty)$ such that $$ {\rm a} +{\rm b}x+\frac{1}{2}x^2{\rm u}=0, \quad \|{\rm u}\|=1 ...
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1answer
21 views

Prove that the second derivative is positive iff the function is convex.

Well, I want to prove the following: Let $f:(a,b)\to\mathbb R$ be double differentiable then $f$ is convex iff $\;\;f''(x)>0$ for all $x\in (a,b)$. Then I tried te following: $\Rightarrow]$ Lets ...
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0answers
14 views

Sobolev space on circle

Let $f : \mathbb{S}^1 \rightarrow \mathbb{R}$ be a function on the circle. I want to calculate the $H^1$ norm of $f$.So I pull this function back to $[0,2 \pi]$ by looking at $f \circ \phi,$ where ...
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1answer
15 views

Bounded second derivative implies square root of f is Lipschitz.

Can you help me with this exercise? Let $f \in C^2(\mathbb{R}) $ a function $ f(x) > 0, \forall x \in \mathbb{R} $ and $\|f''\|_\infty < \infty $ , prove that $\sqrt f$ is Lipschitz ...
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1answer
28 views

How to use Induction with Sequences?

I have posted this similar question here, but with no hopes. I would just like to know: Most of the solution I have no issue with. Look at where they say: "Choose a representation $(n - 3^m)/2 = ...
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1answer
69 views

A curious convergence condition

How do you prove there exists an $M \in \mathbf{R}$ such that $|\frac{-1}{\sin(1)+2} + \frac{1}{\sin(2)+2} +...+ \frac{(-1)^N}{\sin(N)+2}| \le M$ for all $N \ge 1$? This is the conditon for ...
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1answer
22 views

length of parabolic curve

How do you find the length parabolic curve z(t)=t-3it from t=0 to t=1 using the hyperbolic definition of sine and cosine. I am kind of stumped and if I can have a general set up of the problem that ...
2
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1answer
43 views

If x and y are both greater than or equal to 1, show that $|\sqrt{x}-\sqrt{y}|$ is less than or equal to $0.5| x-y |$

If x and y are both greater than or equal to 1, show that $|\sqrt{x}-\sqrt{y}|$ is less than or equal to $0.5| x-y |$ Would really appreciate any help! Thanks
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0answers
14 views

Some inequality related to ODE systems

For $i=1,2$, let $\phi_i:R_+\to R_+$ be a continuous function such that $\phi_i(0)=0$ and define $$\gamma_i(l) := \int_0^l\frac{dm}{\phi_i(m)}.$$ Assume that $(\phi_1,\phi_2)$ satisfy the following ...
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1answer
32 views

Continuous functions spaces

Recently I had to dive into abstract mathematics to understand deeply finite element method (I am an engineer not a mathematician). In some examples of linear spaces it appeared the space: ...
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1answer
37 views

How can i solve $\int_0^t \frac{(t-\tau)^{\frac{1}{2}}}{\tau^{\alpha}}d\tau$,

I want to find the value of the integral $$\int_0^t \frac{(t-\tau)^{\frac{1}{2}}}{\tau^{\alpha}}d\tau,$$ where $0<\alpha<1$. Using Mathematica I found the solution to be ...
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0answers
32 views

Sobolev embedding theorem in the homogeneous case

We know that if $s>\frac{n}{2}$ the following inclusion holds $$H^s(\mathbb{R}^n)\subset L^\infty(\mathbb{R}^n)$$ Is it also true in the case we deal with the homogeneous space ...
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1answer
17 views

Prove that the distance between a disjoint compact set and closed set is nonzero.

Let $X$ be a metric space, and $K$ is compact and $C$ is closed, and $K$ and $C$ are disjoint. Prove that $$\inf_{k\in K, c\in C} d(k,c) > 0$$ What I'm thinking is considering the function $f: K ...
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0answers
22 views

Product of infinite discrete space is second countable

Given $K^w$ equipped with product topology is an infinite product of countably infinite discrete space $K$ . Show that $K^w$ is second countable. My Progress: Since the product topology means there ...
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2answers
53 views

Is it possible to turn infinite sums into infinite products?

I am working on studying infinite products. I know that it is possible to convert an infinite product to an infinite sum using logarithms, where $$\log \prod s_n = \sum \log s_n$$ My question is, it ...
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1answer
42 views

Set and its complement that are both dense

I'm trying to use Baire's theorem to give an example about open sets $\left\{X_i:i\in N\right\}$ in $\Bbb R$ such that $\cap_{i\in N} X_i$ and $\Bbb R - \cap_{i\in N} X_i$ are dense in $\Bbb R$. So ...
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6answers
589 views

In search of a “perfect” test on (positive) series convergence

Thus far mathematicians have developed many powerful tests on the convergence of a positive series (I mean $\displaystyle\sum_{i=1}^{\infty}a_i$ specifically), such as : Cauchy's Testwhich deals with ...
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1answer
25 views

$\lim_{t\to \infty}x(t)$:convergent $\Rightarrow$ $\lim_{t\to \infty}x'(t)=0$

$$\lim_{t\to \infty}x(t)\text{ convergent} \Rightarrow \lim_{t\to \infty}x'(t)=0$$ where $'=\frac{d}{dt}$. Is this proposition the truth?
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0answers
47 views

Let $f:\mathbb{R} \to \mathbb{R}$ monotonic and such that $f(x_1+x_2)=f(x_1)+f(x_2)$, prove that $f(x)=ax$ and $f(1)=a$ [on hold]

Let $f:\mathbb{R} \to \mathbb{R}$ monotonic and such that $f(x_1+x_2)=f(x_1)+f(x_2)$. For all x1 and x2 Real numbers. I want to prove that $f(x)=a\,x$ for all real numbers x where $f(1)=a$.
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0answers
32 views

Proving definition of delta function [on hold]

Consider the function defined $\frac{sin(cx)}{x}$. How would one show that $$\lim_{c \to +\infty} \int_{-\infty}^{+\infty} dx \frac{sin(cx)}{x} f(x) = f(0)$$ for any given function $f(x)$ which shows ...
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0answers
28 views

$A=\{x>0\mid\sum_{n=1}^{\infty}(\sqrt[n]{x}-1)\text{ is convergent}\}$

I have an question , if $$A=\left\{x>0\mid\sum_{n=1}^{\infty}(\sqrt[n]{x}-1)\text{ is convergent}\right\}$$ then determine $A$.
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1answer
51 views

What does $f :[-3,3] \rightarrow \mathbb{R}$ mean?

I don't understand what this means: $f :[-3,3] \rightarrow \mathbb{R}$ defined by $f(x) = 3x^2 - 5x+1$ Does $[-3,3]$ describe an interval?
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5answers
68 views

Starting either Advanced Calculus or Introductory Analysis [on hold]

S.E. advisers, I am a sophomore in US with double majors in mathematics and microbiology; my current computational/mathematical biology research got my interested in the mathematics, particularly the ...
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0answers
20 views

Strichartz estimates for wave equations

Let's consider the wave equation $\Box u=F $ with $u(0)=g_0$ and $\partial_t g(0)=g_1$. Strichartz estimates tell us that $$\Vert u\Vert_{L^q_tL^r_x}+\Vert u\Vert_{C^0_t\dot{H}_x^s}+\Vert\partial_t ...
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0answers
12 views

Is such a function integrable?

Let $f$ be a real valued $2\pi$ periodic function on $\mathbb R$, integrable on compact intervals. Let $f$ be differentiable at $x_0$. We define $$ g(x)=\frac{f(x_0+x)-f(x_0-x)}{2 \sin x} $$ for ...
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5answers
30 views

Prove that $m$ is an integer

Suppose $n$ is a odd integer. It satisfies: $$3^{s} < n < 3^{s+1}$$ For some integer $s \ge 0.$ Show that: $$m = \frac{n - 3^{s}}{2}$$ Is an integer. So, $$2m = n - 3^{s}$$ But that wont ...
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2answers
58 views

Can we describe all group isomorphisms from $(\mathbb R ,+)$ to $(\mathbb R^+ , .)$ ?

Can we describe all group isomorphisms from $(\mathbb R ,+)$ to $(\mathbb R^+ , .)$ ? I have tried that if $f$ is such an isomorphism , then $f(x)>0$ , and $f(r)=(f(1))^r , \forall r \in \mathbb Q$ ...
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1answer
28 views

If $I$ is a proper ideal of $C[0,1]$ , then should there exist $a \in [0,1]$ such that $f(a)=0 , \forall f \in I$?

Let $C[0,1]$ be the ring of all real valued continuous functions under point-wise addition and multiplication . We know that for every $a \in [0,1]$ , $\{f \in C[0,1] : f(a)=0\}$ is a proper ideal of ...
2
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0answers
30 views

Functions so that image of min (resp. max) is a positive definite kernel

I am trying to determine the functions $\phi : \mathbb{R}^+ \to \mathbb{R}$ such that: Pb 1: $K(s, t) = \phi( \mathrm{min} (s,t))$ is a positive definite kernel on $\mathbb{R}^+$. Pb 2: $K(s, t) = ...
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1answer
18 views

Mapping properties of the complex inversion $1/z$

This is from the text Complex Variables with Applications by Herb Silverman. We present some precise mapping properties of $w=1/z$. Consider the circle $|z-a|=R, a\neq 0$. If $w=1/z$, then we obtain ...
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6answers
2k views

Are there many fewer rational numbers than reals?

Today my professor asked me to figure out the probability of getting a rational number from $[0,1]$. His answer was that the probability is $0$. Why is this?
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1answer
27 views

If $f$ and $g$ have the same $L^{2}$ norm, does it imply $hf$ and $hg$ have the smae $L^{2}$ norm?

Let $f,g \in L^{2}(\mathbb R)$ with $\|f\|_{L^{2}}=\|g\|_{L^{2}}$( that is, $f$ and $g$ have the same $L^{2}-$ norm). We choose $h\in \mathcal{S}(\mathbb R)$(= Schwartz space) so that $hf, hg \in ...
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2answers
47 views

Show that function $f(a,b)$ is differentiable

Show that function $f: \mathbb{R}^2 \rightarrow \mathbb{R} $ $f(a,b) = \dfrac{a^3+b^3}{\sqrt{a^2+b^2}}$ if $(a,b)\in\mathbb{R}^2 -\{(0,0)\} $ and $0$ if $(a,b)=(0,0)$ is differentiable on ...
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2answers
35 views

Calculus by Apostol Exercise 1.26 number 26

What I did first is to use theorem 1.18 to subtract a from the limits of integration of the integral of f(x), leaving the new limits of integration to be from 0 to (b-a). In that case I can use ...
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1answer
47 views

Problem 5 in Exercises following Sec. 16 in Munkres' TOPOLOGY, 2nd edition [on hold]

Let $X$ and $X^\prime$ denote a single set in the topologies $\mathscr{T}$ and $\mathscr{T}^\prime$, respectively; let $Y$ and $Y^\prime$ denote a single set in the topologies $\mathscr{U}$ and ...
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1answer
26 views

Convergence of a sequence of a decreasing family of compact sets.

So I'm given a decreasing family of compact sets $(K_n)$ in $\mathbb{R}$ such that $K_1\supset K_2\supset K_3\supset \cdots$ and have to show that for a sequence $(a_n)$ such that $a_n \in K_n$ there ...
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1answer
17 views

Proof of a corollary of the Banach Fixed Point Theorem

If $(X,d)$ is a complete metric space, and $f: X \rightarrow X$ is a continuous function, show that if $f^{N}$ is a contraction (for some $N > 0$),then $\exists! x \in X$ such that $f(x) = x$. I ...
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3answers
53 views

Is $\int_a^{b}f(x) dx = \lim_{k\rightarrow \infty } \int_a^{b_k}f(x)$?

Let $f:[a,b] \rightarrow \mathbb{R}$ be integrable. Let $b_k \subset [a,b]$ be a sequence such that $\lim_{k\rightarrow \infty} {b_k} = b$ Consider the integral $\int_a^{b}f(x) dx$. Is it always the ...
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0answers
13 views

Why is $\ell_{2,1}$ mixed norm non-smooth?

I'm reading about optimization problems involving mixed norms. In particular I'm getting acquainted with the $\ell_{2,1}$ norm. For a matrix $\mathbf{X}$, the $\ell_{\alpha,\beta}$ norm, ...
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0answers
28 views

How do I prove the following WITHOUT using the open mapping theorem?

I want to prove the following claim: Claim: Suppose $H$ and $K$ are Hilbert spaces. Let $T$ be a bounded and injective linear operator from $H$ to $K$. If the range of $T$ is closed, then there is a ...
9
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1answer
78 views

If $\sum_{n=1}^\infty\vert a_n\sin(nx)\vert$ converges, then $\sum_{n=1}^{\infty}\vert a_n\vert<\infty$.

Suppose $\sum_{n=1}^\infty\vert a_n\sin(nx)\vert$ converges for all $x$ in a set of positive measure $A$. I'm trying to prove $\sum_{n=1}^{\infty}\vert a_n\vert<\infty$. The only useful result I ...
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0answers
61 views

If $f(x+y)=f(x)+f(y)$ in real numbers, does the function $f$ have to be continuous? [duplicate]

Given $f:\mathbb{R}\rightarrow\mathbb{R}$, and the function $f$ satisfies $f(x+y)=f(x)+f(y)$ for any $x,y\in S$. Can we say that this function $f$ must be continuous? I think it is false, but ...
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3answers
50 views

Calculate $\lim_{x \to -1}(\frac{x+1}{x-1})^x$, $\lim_{x \to 1}(\frac{x+1}{x-1})^x$, and $\lim_{x \to \pm \infty}(\frac{x+1}{x-1})^x$ [on hold]

Why are the following limit $+\infty$? $$\lim_{x \to -1}(\frac{x+1}{x-1})^x$$ $$\lim_{x \to 1}(\frac{x+1}{x-1})^x$$ Also, how do I find $$\lim_{x \to \pm \infty}(\frac{x+1}{x-1})^x?$$
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2answers
38 views

Study of the first and second derivative of $\sqrt{|x^2+x|}-x$

I am not able to study the positivity of the first and second derivatives of $\sqrt{|x^2+x|}-x$ (that is, the values of $x$ for which the derivatives are positive, negative, or zero), because the ...
1
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1answer
23 views

Show that $\langle \phi(x) - \phi(y), x-y \rangle \ge c|x-y|^2$ and $|\phi(x)- \phi(y)| \ge |x-y|$.

Let $\phi: \mathbb{R}^n \rightarrow \mathbb{R}^n$ a function $C^1$. Suppose there is a constant $c>0$ such that $$\langle \phi'(x)h,h \rangle \ge c |h|^2, $$ for all $h \in \mathbb{R}^n$. Show ...
1
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2answers
57 views

Computing $\lim_{(x,y)\to (0,0)}\frac{\sin(x+y)}{x+y}$

I'm trying to compute the following limits and the textbook that I'm looking at suggested the following method. $$\lim_{(x,y)\to (0,0)}\frac{\sin(x+y)}{x+y}$$ $$\lim_{(x,y)\to ...