0
votes
1answer
11 views

Countable and uncountable sets.

a) Show that $\left \{ n^{2}+m^{2}:n,m\in \mathbb{N} \right \}$ is countable. b) Show that $\left \{ x\in \mathbb{R}:x(x-2)<0 \right \}$ is uncountable. My answers: a) Is it possible to define ...
0
votes
1answer
8 views

Determine $\left \{ u\geq a \right \}$ for all $a\in \mathbb{R}$, and is $u$ $\mathcal B(\mathbb{R})/\mathcal B(\mathbb{R})$-measurable?

Let $u:\mathbb{R}\to\mathbb{R}$ be given by $u(x)=\left \lfloor x \right \rfloor$. Determine the set $\left \{ u\geq a \right \}$ for all $a\in \mathbb{R}$. Show that $u$ is $\mathcal ...
1
vote
1answer
13 views

Compact operators, space of sequences

Let $\phi\in\ell^\infty$. For $p\in[1,\infty]$, define $M_\phi:\ell^p\to\ell^p$ by $$M_\phi(f)=\phi f.$$ Show that $\Vert M_\phi\Vert=\Vert\phi\Vert_\infty$, and $M_\phi$ is compact if and only if ...
1
vote
0answers
17 views

Is the space $B([a,b])$ separable?

Let $a$, $b$ be two real numbers such that $a < b$, and let $B([a,b])$ denote the metric space consisting of all (real or complex-valued) functions $x=x(t)$, $y=y(t)$ that are bounded on the closed ...
0
votes
2answers
20 views

Use mathematical induction to prove Σ n,k=1 (1/k(k+1)) = (n/n+1) for all n in Natural numbers?

This is how far I can get: p(n): nΣk=1 (1/k(k+1)) = (n/n+1) p(1): 1Σk=1 (1/(1+1)) = (1/1+1) => 1/2 = 1/2 p(1) is true. Assume that p(k) is true. p(k) = kΣk=1, (1/k(k+1)) = k/k+1 Show ...
1
vote
1answer
25 views

Polar Coordinates in $\mathbb R^n$

After proving Fubini-Tonelli theorem a formula on polar coordinates in $\mathbb R^n$ is given in my class as follows. Let $f$ be a real-valued integrable function on $\mathbb R^n$ and $S^{n-1}$ be the ...
0
votes
2answers
28 views

Fixed point in compact metric space

I guys! I try to solve the following small problem. However, I'm not able to prove the second part. In particular, I have some problems in using the compactness hypothesis on $X$ to find proper ...
0
votes
1answer
20 views

Advanced Calculus Question. Prove (sn + tn) is a Cauchy sequence

Based on the definition of a Cauchy sequence, that if (sn) is a Cauchy sequence and (tn) is a Cauchy sequence, then (sn + tn) is a Cauchy sequence I try to work from the definition that |(Sn + tn) ...
1
vote
1answer
26 views

Continuous functions unbounded on set

For Each of the sets construct a continuous function that is unbounded on the set. $\Bbb N$ $(2,3)$ $\left\{\frac 1 n \mid n \in \Bbb N\right\}$ $[0, \sqrt 2]\cap \Bbb Q$ ...
0
votes
1answer
26 views

Show that $\int_{X}u\, \mathrm{d}\mu\leq 4$ and $\int_{X}u\, \mathrm{d}\mu=1$.

Let $(X,\mathcal{A},\mu)$ be a measureable space. Let $u\in \mathcal{M}_{\mathbb{R}}^{+}(\mathcal{A})$ and $\lbrace u_{j}\rbrace_{j\geq 1}$ be a sequence of functions in ...
0
votes
0answers
24 views

Find the pointwise limit of {gn} on [0, ∞). Please help!!

Consider sequence of functions gn(x)=x\over 1+x^n over [0,\infty) (a) Find the pointwise limit of {gn} on [0, ∞). g(x)= x 0 \le x \lt x 1/2 x=1 00 x>1 Show gn(x) ...
3
votes
1answer
25 views

First order PDE with discontinuous coefficients

I want to consider the following equation $$u_t+\mathrm{sgn}(x)u_x=0,\,\,u(0,x)=u_0(x)$$ Now if $x>0$ or $x<0$ I can use the method of characteristics to obtain $u(t,x)=u_0(x-t)$ if $x>t$ and ...
1
vote
0answers
26 views

Strange inequality

I found the inequality $\beta e - \frac{3}{2} n \ log(e+Bn)+ \frac{5}{2} \ n \ log(n) + const \cdot n \geq \frac{\beta e}{2}+ \beta n $ in a textbook,provided that either $e$ or $n$ is large. We ...
2
votes
1answer
36 views

Prove that this function is Borel measurable

Prove that if $s\ge 0$, $f:\mathbb{R}^n\to\mathbb{R}^m$ is continuous and $K\subset\mathbb{R}^n$ is compact, then the function $$ F:\mathbb{R}^m\to [0,\infty]\\y\mapsto H^{s}(K\cap f^{-1}(\{y\})) $$ ...
0
votes
0answers
24 views

Determine integrals $\int_{\mathbb{R}}u\, \mathrm{d}\delta_{3}$ and $\int_{\mathbb{R}}u\, \mathrm{d}\delta_{\pi}$.

Consider the function $u:\mathbb{R}\to [0,\infty]$ given by $$ u(x)=\sum_{n=1}^{\infty}\frac{1}{n^{2}}1_{[n,n+1]}(x) $$ I have determined that $\int_{\mathbb{R}}u\, \mathbb{d}\lambda=\pi^{2}/6$ where ...
1
vote
0answers
45 views

Ode with step function in the right-hand-side

I want to solve the following ODE: $$\dot{X}(t,x)=F(X(t,x))$$ where $F(x)=1$ if $x>0$ and $-1$ if $x<0$. How to treat this discontinuous right-hand-side?
0
votes
1answer
61 views

Prove (or disprove) this $\sum_{n=1}^{\infty}\dfrac{a_{n}}{n}$ is convergence? [on hold]

Let $a_{n}>0$ be a sequence, and $0<a\le 1$, such that $\sum\limits_{n=1}^{\infty}(a_{n})^a$ converges. Prove or disprove $\sum\limits_{n=1}^{\infty}\dfrac{a_{n}}{n}$ is convergent. I ...
1
vote
2answers
26 views

Time derivative of operator

I have to compute, at least formally, the following derivative $$\partial_t \exp(it\Delta)f(x-ct)$$ where $\Delta$ is the Laplacian and $c$ is a constant. I know that $e^{it\Delta}$ is the Schrodinger ...
0
votes
0answers
14 views

Theorem about n=1 wave equation in Evans

In Evans, PDE edition 2 on p68 we have a Theorem that tells us some properties about the solution to the wave equation for $n=1$. It reads: Assume $g \in C^2(\mathbb{R})$, $h\in C^{1}(\mathbb{R})$, ...
0
votes
1answer
15 views

Correction of Proof that if f:[0,1] is a continuous function and f(x)>2 with x being in [0,1) it is not necessary that f(1)>2

Here's how I wrote it up: To approach this consider an example of a continuous function which fails to satisfy f(1)>2 even though it satisfies f(x)>2 for x in [0,1). My counterexample is a negative ...
1
vote
0answers
30 views

Method of characteristic for second order pde

Can I use the method of characteristic to solve second order pdes? For instance I canconsider the equation $$u_t+u_x=u_{xx}$$
0
votes
1answer
30 views

Sequence of functions that converges a.e. but not in the $L^1$ norm

How can I construct a sequence of functions that converges a.e. but it does not in the $L^1$ norm?
0
votes
2answers
33 views

$f(x)=1$, for every $x \in [0,1]$ if $f:[0,1]\to\mathbb R$ is continuous and $f(p)=1$ for every $p\in [0,1]\cap\mathbb Q$.

How would you approach this if I have to use the fact that "every number is a sequence of rational numbers"? Currently, I am proving this by contradiction in the following way: Let f(p)=1 for all ...
0
votes
0answers
29 views

Tricky Change Of variables?

If $f(x,y,z)$ is a differentiable function then from $\mathbb{R}^3$ to $\mathbb{R}$ then: if $w_1:=x+y$, $w_2:=\frac{x-y}{x+y}$ and $w_3:=x^2+y^2+z^2$ then what would the function resulting from a ...
1
vote
1answer
29 views

$k^m$ where $k$ is infimum of $\{x \in \mathbb{Q} | m \le x^m\}$

if $k = \inf\{x \in \mathbb{Q} | m \le x^m\}$ can I say that $k^m = m$? Can you please explain why yes or why not? Any help is appreciated. Thank you
0
votes
0answers
8 views

Show that $ f$ is strongly differentiable at $x_0$ .

Definition: Let $U\subseteq \Bbb R^m$ be an open set. Let $f: U \to \Bbb R^n$ be a function and $T: \Bbb R^m \to \Bbb R^n$ be a linear transformation. We say that f is strongly differentiable ...
0
votes
1answer
24 views

Let $A$ be an infinitely countable set and let $B$ be a finite set. Show that $A \cup B$ is also countable.

Let $A$ be an infinitely countable set an let $B$ be a finite set. Show that $A \cup B$ is also countable. In the solution for this exercise, a function $f: \mathbb{N} \to A \cup \left ( B \setminus ...
0
votes
0answers
13 views

Show that if f is strongly differentiable at $x_0$ then it satisfies Lipschitz condition in a neighbourhood of $x_0$.

Definition: Let $U\subseteq \Bbb R^m$ be an open set. Let $f: U \to \Bbb R^n$ be a function and $T: \Bbb R^m \to \Bbb R^n$ be a linear transformation. We say that $f$ is strongly differentiable ...
1
vote
2answers
35 views

Show that there exists a subsequence $\{F_{n_{k}}\}$ which converges to uniformly on $[a,b]$.

Let $\{f_n\}$ be uniformly bounded sequence of functions which are Riemann-integrable functions on $[a,b]$ and define for $a\leq x\leq b$. $$ F_n(x)= \int_a^x f_n(t)dt.$$ Show that there exists a ...
0
votes
0answers
37 views

If f is a continuous function on $[0,1]$ and if for each $n=0,1,2,3,…$. Prove that $f=0$ on $[0,1]$.

If f is a continuous function on $[0,1]$ and if for each $n=0,1,2,3,...$ $$ \int_0^1 f(t)t^n dt=0$$ Then prove that $f=0$ on $[0,1]$. Here I don't want the proof. I have one proof. But I have ...
0
votes
0answers
21 views

If a real valued function is increasing bounded above, does that imply that $\lim_{x\to \infty}f(x)$ exists?

I have a quick question: If a real valued function is increasing on $[x_0,\infty)$ and bounded above, does that imply that $\lim_{x\to \infty}f(x)$ exists ? I know that if a sequence is increasing ...
2
votes
1answer
32 views

Which negation of the definition of a null sequence is correct?

A Cauchy sequence $a_n$ is said to be a null sequence if for every $\varepsilon>0,$ there exists an integer $N$ s.t. $\forall n >N, \lvert a_n \rvert < \varepsilon$. I thought the negation ...
2
votes
0answers
44 views

Sign of a Function over $[0,\pi]$

Let $\lambda \in \mathbb{R} $ such that $\left| \lambda \right| \ne 1$ $$f_\lambda (x)=(λ\cos x−1)(λ−\cos x)$$ Show that : For $|λ|<1$, $$f_λ(x)=\begin{cases} <0 ...
1
vote
4answers
55 views

How to give a rigorous proof of this fact about closures of open balls in the euclidean spaces?

Let $n$ be a positive integer, $\vec{a} \in \mathbb{R}^n$, and $r > 0$. Then it is intuitively clear that the closuer of the open ball $$B(\vec{a} ; r) \colon= \{ \vec{x} \in \mathbb{R}^n \colon ...
1
vote
2answers
37 views

Inequality for all real numbers

I am trying to prove a hw problem from Taos Analysis 1 book. I would like some help proving the following statements if they are true which I do not necessarily believe. Let $x,y \in \Bbb R$. Show ...
0
votes
1answer
43 views

(absolute) Convergence of a series

I want to prove that the following series is convergent for $x>0$: $$ \sum_{n=1}^\infty \left( \prod_{p\mid n} \frac{1}{p-1}\right) n^{-x} $$ I tried to estimate the product but I didn't get so ...
1
vote
2answers
31 views

$C^n[a,b]$ as a normed algebra

I would like to prove that the space of complex valued functions, differentiable $n$ times with continuous derivative, $C^n[a,b]$, with the metric defined by the norm ...
3
votes
1answer
23 views

If $f$ is equal to an affine function up to $1$-th order at $a$, then $f$ is differentiable at $a$, proof more subtle then it appears?

I came across the following exercise: Two functions $f, g : \mathbb R \to \mathbb R$ are equal up to $n$th order at $a$ if $$ \lim_{h \to 0} \frac{f(a + h) - g(a + h)}{h^n} = 0. $$ Show that $f$ ...
0
votes
1answer
27 views

What does monotonically convergent mean in this example.

Suppose that $$f(x)=\sum_{n=1}^{\infty}f_n(x)\,\,\,\,\,\,\,\,\,\,(x\in X)$$Where $f_n:X\rightarrow [0,\infty]$ for $n=1,2,3...$ Let $g_N=f_1+...+f_N$. Then the sequence $\{g_N\}$ converges ...
0
votes
2answers
33 views

How to prove that a function is left continuous

I cannot work out this problem even though it seems not that difficult. Could anyone kindly give me any hint? Thanks! If $f(x)$ is measurable on $E \subset \mathbb R$, then $$ \varphi (t)=m\big(\{x ...
1
vote
2answers
51 views

Derivatives vanishing at infinity

Take a subinterval of Euclidean space, for instance, which has infinite length. WLOG let it be $\mathbb{R}$. Is there an example of a function $f$ such that $f$ and $f''$ vanish at infinity, but not ...
2
votes
1answer
29 views

Closure and subbasis

Let $X$ be a topological space and $A \subset X$ with a subbasis $S$. Does it then hold that $x \in \overline{A}: \Leftrightarrow \forall s \in S: (x \in s \Rightarrow s \cap A \neq \emptyset).$ This ...
-2
votes
1answer
28 views

On Pseudometric

How a pseudometrics induces topology? Can anyone discuss on this topic or give any good reference?
1
vote
1answer
47 views

If $f$ is differentiable everywhere, is $f'$ the weak derivative?

Let $f \in C^0([0,T])$ be such that $f'$ exists in the classical sense everywhere, but $f'$ may not be continuous. Is it true that $f'$ is the weak derivative of $f$ too, if it exists? I know this is ...
0
votes
1answer
29 views

Hölder norm bounded by $L^p-$norm?

Let $C_0^{\alpha}(\mathbb{R})$, $0<\alpha<1$ denote the space of Hölder-continuous functions on $\mathbb{R}$ with compact support. Is it true that for any $f\in C_b^{\alpha}(\mathbb{R})$ one ...
0
votes
2answers
51 views

Subject GRE question - set of points of discontinuity

I was just working on a Math Subject GRE practice test, and I got the following problem wrong: Let $f$ be the function defined on the real line by $\displaystyle f(x) = \begin{cases} \displaystyle ...
1
vote
0answers
39 views

Bounded variation in the context of Feller's paper on Muntz' Theorem

The paper I have posted a picture of is a paper of Feller. He shows that the functions $f_k$ are Laplace transforms of $C^\infty$ functions $u_k$. In order to execute his suggested proof, I ...
2
votes
1answer
24 views

Is every Lipschitz continuous function is holder continuous with exponent $\in (0,1)$?

Is every Lipschitz continuous function is holder continuous with exponent $\in (0,1)$? This seems to be true,but I haven't found such a conclusion in any textbook.
2
votes
1answer
26 views

Solution transport equation

I have to solve the following equation $$\partial_t v+2A\cdot\nabla v-iB(x)\cdot Av=0$$ where $A$ is a constant vector and $B$ a smooth vector field. I can solve the transport equation $\partial_t ...
0
votes
1answer
44 views

Whether there is a continuous bijection from $(0,1)$ to closed interval $[0,1]$. [duplicate]

Is there a continuous bijection from open interval $(0,1)$ to $[0,1]$. The answer is not. How to prove? I think it may proceed by contradiction and apply open mapping theorem. However, $(0,1)$ is ...