0
votes
1answer
11 views

Frechet derivative of the sup norm function on $C[0,1]$

I know that the derivative in the title does not exist for any x but I do not have a clue why. Could someone explain why this derivative DNE at 0 and I can figure out why it does not exist at x? ...
0
votes
0answers
13 views

Biorthogonal functionals continuous?

If I have a Schauder basis $(x_n)$ of a Banach space $X$. Such that for every $x = \sum_{i=1}^{\infty} a_i x_i$ for a unique sequence $(a_i) \subset \mathbb{R}$. Is it obvious that the functionals ...
1
vote
0answers
21 views

question about a proof: sequence of picard iteration converges uniformly

Given $u'(t)=t\cdot u(t)+t^3$ with $u(0)=0$ I want to show that $$u_{n+1}(t)=u(t_{0})+\int_{t_0}^{t}f(x,u_{n}(x))dx$$ converges uniformly on $[-b;b]$ and solves the differential equation. After ...
0
votes
0answers
20 views

If $f_{n} \in L^{\infty}$, $ \int_{0}^{1}f_{n_{k}}(x)g(x)dx \rightarrow \int_{0}^{1}f(x)g(x)dx$ for every $g \in L^1$

Supposet that $\{f_{n}\}_{n=1}^{\infty} \in L^{\infty}$. Is the following statement always true? There is a subsequence $\{n_{k}\}$ and a function $f \in L^{\infty}$ such that $$ ...
0
votes
1answer
33 views

solve the functional equation

Let $\phi : R-> C $ (complex numbers) $\phi(0)=1$ $ \phi(-t) = \overline{\phi(t)} $ ( continuous and bounded) solve the functional equation: $Re \phi(t)= \phi(t) \overline{\phi(t)}$ This is all ...
1
vote
0answers
28 views

Radius of convergence and uniqueness

I was reviewing some fundamental concepts of real analysis, and the following exercise popped up: Let there be a power series: $\sum\limits_{k=0}^\infty a_k \mid x - 3 \mid $ Such that for $\ x = ...
1
vote
1answer
24 views

Prove $e^c>c^e$ if $c>0$ and $e \neq c$ using graph.

I am on this question where it tells me to show $e^c>c^e$ if $c>0$ and $e \neq c$ using the graph of $\dfrac{(log(x))}{x}$. Now it is obvious that the graph reaches a maximum at $x=e$ but how ...
1
vote
1answer
28 views

find the maximum of the function F under the condition $ \sum_{i=1}^N x_i = 1$

Let F a function of $ \mathbb{R} ^N_+ \rightarrow \mathbb{R}$ defined as : $$F(x_1,..,x_N)= - \sum_{i=1}^N x_i log(x_i) , x_i \gt 0$$ How can i find the maximum of the function F under the ...
2
votes
0answers
24 views

Lower bound of $\sum_{k = 1}^{N}1/(x + k)$

Let $f(x) := \sum_{k = 1}^{N}1/|x + k|$ for $x \in [0, N]$. Why is $f(x) \geq C\log N$ for all $x \in [0, N]$ where $C$ is an absolute constant. My work is: Since $x \in [0, N]$, we can remove the ...
2
votes
1answer
36 views

Estimate for weak $L^{1}$ norm

Let the weak $L^{1}$ norm on $f$ be defined by $\|f\|_{\mathrm{WL}^{1}} = \sup_{t > 0}t D_{f}(t)$ where $D_{f}(t) = \mu(\{x \in \mathbb{R}: |f(x)| > t\})$ and $\mu$ is the standard Lebesgue ...
1
vote
1answer
12 views

Is the composition of monotone operators monotone?

Let $H$ be a real Hilbert space with inner product $\langle\cdot, \cdot \rangle: H \times H \rightarrow \mathbb{R}$, and induced norm $\left\| \cdot \right\|: H \rightarrow \mathbb{R}_{\geq 0}$. Let ...
6
votes
3answers
45 views

Counterexamples for Hölder's inequality when $p$ and $q$ are not conjugate.

Hölder's inequality shows that, when $$ \frac{1}{p} + \frac{1}{q} = 1,$$ and $f\in L^p$ and $g\in L^q$, then $$\Vert f\,g\Vert_1 \le \Vert f \Vert_p \Vert g \Vert_q.$$ Is there an example of this ...
-3
votes
1answer
21 views

define convergence for a sequence of elements in a metric space [on hold]

Define the convergence for a sequence of real numbers. How will you modify the definition to define convergence for a sequence of elements in a metric space? Give examples of convergent and non ...
2
votes
2answers
77 views

If $f$ is twice differentiable then $f^{-1}$ is twice differentiable

$f:(a,b) \rightarrow (c,d)$ is a bijection and $f$ is differentibale with $f'(x) \neq 0$ for all $x \in (a,b)$, then $f^{-1}$ is also everywhere differentiable. Show that if $f$ is twice ...
0
votes
1answer
24 views

Continuity of the operator

Let $D: (C^1[a,b], \Vert .\Vert_1 )\rightarrow (C[a,b], \Vert .\Vert_1)$ with $D(f)=f^{\prime}$ I wonder if I will be continuing this operator Note: $\Vert f\Vert_1=\int_a^b\vert f(x)\vert dx$ All ...
0
votes
0answers
16 views

Prove that $h(f(x,y))$ and $h(g(x,y))$ are first order approximations at $(x_0, y_0)$ or find a counter example

Let $f: \Bbb R^2 \to \Bbb R$ and $g: \Bbb R^2 \to \Bbb R$ be first order approximations at $(x_0, y_0)$, and let $h: \Bbb R^2 \to \Bbb R$ be continuous. Prove that $h(f(x,y))$ and $h(g(x,y))$ are ...
2
votes
1answer
25 views

Hyperbolic Systems ODE

Let $M_n$ the set of matrices of order $n \times n$ identified with $\mathbb{R^{n^2}}$ e $S=\{A \in M_n ; x'=Ax$ is hyperbolic$\}$. Show that $S$ is open and dense $M_n$.
3
votes
1answer
53 views

On the equation $\exp(a x+b)=\ln(x)$

I am confronted with: $$\exp(a x+b)=\ln(x)$$ for $a,b$ reals and $a<0$, $b>0$. I need the (unique) solution for $x$. My first target is (if it exists) an analytic solution in terms of ...
0
votes
3answers
40 views

Proving Differentiability rigorously

Assume that real function f is differentiable at $x_0$ with $f'(x_0)$ >0. How would one show that there exists a $\delta$>0 such that $$ f(x)>f(x_0) $$ for all x in between $x_0$ and $ x_0 + ...
2
votes
1answer
23 views

Radius of Convergence of Sum of two Series.

Hi all, I know there are similar questions on here, but none deal with the fact of trying to prove that $T \geq min\{R,S\}$. Intuitively this doesn't make sense to me, If you have, ...
1
vote
1answer
23 views

Question involving intermediate value theorem

The function $g: (0, \infty) \to \mathbb{R}$, is continuous, $g(1)>0$ and $$\lim_{x \to \infty} g(x) = 0$$ It is a fact that for every $y$ between $0$ and $g(1)$ the function takes on a value in ...
3
votes
4answers
115 views

Is $\sum_{n=1}^\infty {1\over 3^{\sqrt{n}}}$ convergent?

Is $\sum_{n=1}^\infty {1\over 3^{\sqrt{n}}}$ convergent ? I use it to compare with $1/n^2$, and then I used LHôpitals rule multiple times. Finally , I can solve it. However,I think we have other ...
0
votes
1answer
41 views

Criteria to be in weak $L^{p}$ space

Let $X$ be a $\sigma$-finite measure space. Let $f : X \rightarrow \mathbb{C}$ be a measurable function and $1 < p < \infty$. Suppose for $f$ there is a constant $C$ such that ...
1
vote
1answer
40 views

a question about convergence of sequecce!I have tried cauchy method, but it doesn't work

suppose $a_n>0$,and$\sum_{i=0}^\infty a_i$ is convergent,so we need to prove $\sum_{n=1}^\infty{ {1\over n}(a_n+a_{n+1}+\cdots+a_{2n})}$ is also convergent! I have tried cauchy method, but maybe ...
4
votes
1answer
55 views

Calculus and infinitesimals

In the definition of reimann integral, why do we put a 'dx' inside the integral sign when practically it serves no purpose except maybe telling what variable you are talking about. Then in some ...
1
vote
1answer
31 views

Pointwise convergence of arctan(nx)

Question 6 section 8.1 of Introduction to real analysis by Bartle and Sherbert. Show that lim(Arctan nx) = (pi/2)sgn x for x in R, x>=0. I have a final coming up and I've started doing some of the ...
0
votes
0answers
26 views

Intermediate value of the derivative.

Hi all what would the best way be to approach this question? I tried using the hint but I can't seem to formulate an answer for the fist part. Any help for the first and second parts of the question ...
0
votes
1answer
24 views

Minimisation of Finite sum of a decreasing sequence

If $a_{1}<a_{2}<a_{3}<...<a_{n}$, find the minimum value of $$\sum_{i=1}^{n} (x-a_i)^{2}$$ Then find the value of $$f(x)=\sum_{i=1}^{n} |x-a_i|$$ Hi all, what would the best way be ...
0
votes
1answer
27 views

$C^{-1} (1+|x|^{2})^{\frac{s}{2}} \leq (1+|x|)^{\frac{s}{2}} \leq C (1+|x|^{2})^{\frac{s}{2}}$?

Let $s\in \mathbb R,$ and define $f: \mathbb R^{n}\to [0, \infty)$ such that $f(x)= (1+|x|^{2})^{\frac{s}{2}}, (x\in \mathbb R^{n})$ and $g:\mathbb R^{n}\to [0, \infty)$ such that $g(x)= ...
2
votes
1answer
39 views

Del on Riemannian manifolds

I was supposed to figure out what $grad(div(e_r))$ is, where $e_r$ is the unit vector in spherical coordinates. In the following I assume $g:=diag(1,r^2,r^2sin^2(\theta))$ be the metric tensor. My ...
0
votes
0answers
26 views

Find an analytic function [duplicate]

Is it possible to find an analytic expression for a smooth, continuous, single-variable function $y=f(x)$ such that: 1) $f(0) = 0$ 2) $f(x_1) = y_1$ 3) $f(x) > 0, \forall x>0$ 4) ...
1
vote
1answer
27 views

differential equation as taylor series

Consider the equation $\frac{\mathrm{d} x(t)}{\mathrm{d}t} = g(x(t))$ , with $x(0) = x_0$, where g is function that admits derivatives of all orders.If the solution of the equation can be written as a ...
6
votes
4answers
79 views

A question about inequality ${(n+1)\over e^n}^n<n!$

How to prove the inequality $${(n+1)\over e^n}^n<n!$$ I have tried mathematical induction, but it doesn't work! Are there other methods to solve it?
0
votes
1answer
27 views

Prove that the convergence of the sequence (sn) implies the convergence of (s3n) [on hold]

Case 1: $s>0$. Assume $s>0$. Then there exists $N$ such that for all $n>N$ $s_n>0$. If $s_n^3-s^3$ converges to $0$, then write $s_n-s$, as was done in OH, as ...
3
votes
1answer
33 views

Is there a subset of R such that their Cantor-Bendixson rank is the first limit ordinal?

I'm looking for a set $A \subset \mathbb{R}$ such that $\bigcap^\infty_{n=0} A^{(n)} $ is a perfect set (i.e $X'=X$) but $\forall n \in \mathbb{N}$ the set $A^{(n)}$ isn't perfect (where ...
-4
votes
0answers
22 views

let E=C[X] be a normed space and T∈ L(E)… prove that.. [on hold]

Let E=C[X] be a normed space and T∈ L(E). And let $$\||P||_\ = \left\{ \sum ||P^{(n)}||_\infty, \; \; 0 \leq n \leq ∞ \right\}.$$ where $\||P||_\infty$=sup|p(x)|, 0≤x≤1 1- Justify that T:E→E ...
3
votes
0answers
35 views

Fundamental Theorem of Calculus and inverse..

If $F(x)$ is defined as $$F(x)= \int_{a}^{x} f(t) dt$$ calculate $(F^{-1})'(y)$ in terms of $f$. I have been working on this for a while now, does the aanswer to this incorporate the Inverse ...
0
votes
1answer
17 views

Nonexpansive Affine Operators in Hilbert spaces

Let $H$ be a Hilbert space with real inner product $\langle \cdot, \cdot \rangle : H \times H \rightarrow \mathbb{R}$. Let $T$ be an affine operator. Show that $T$ is nonexpansive, i.e., $\left\| ...
0
votes
1answer
38 views

Weierstrass Caratheodory on open interval

I have been working on this question for a while now, and if I have understood it correctly shouldn't the answer be that $\phi_{c}=f'(x)$ for all $x \in (a,b)$ as the function f , is now said to be ...
0
votes
1answer
28 views

How to interchange sum and integral when measure is in terms of Dirac measure?

Let $\{c_{k}\}_{k\in \mathbb Z} \subset \mathbb C$ such that, $\sum_{k\in \mathbb Z} |c_{k}| < \infty.$ Let $\delta_{k}$ is the unit Dirac mass at $k $, we note that $\mu = \sum_{k\in \mathbb Z} ...
1
vote
1answer
40 views

What is the limit of this function as $(x,y)$ approaches $(0,0)$?

Let the function $f \colon (\mathbf{R}^2 \setminus \{(x,y) \in \mathbf{R}^2 \colon x+y = 0 \}) \to \mathbf{R}$ be defined as follows: $$ f(x,y) \colon= \frac{xy}{x+y}$$ if $(x,y) \in \mathbf{R}^2$ ...
0
votes
2answers
24 views

How to quantify this statement

How do I quantify this statement? Let $x \in \mathbb{R}$. $1 \le x \le 2$ if and only if $1 \le x \le 1+ 1/n$ for some $n \in \mathbb{N}$. When I am trying to prove this, I am led (in the course of ...
7
votes
0answers
92 views

Problem 6 - IMO 1985

For every real number $x_1$ construct the sequence $x_1,x_2,x_3,\ldots$ by setting $x_{n+1}=x_n(x_n+\frac{1}{n})$ for each $n \ge 1$. Prove that there exists exactly one value of $x_1$ for which $0 ...
1
vote
3answers
21 views

Negation of continuity applied to a sequence

Show that if it is not true that $\lim_{x \to a} f(x)=l$ then $\exists$ $\epsilon$>0 and a sequence $(x_{n}) \rightarrow a$ as $n \rightarrow \infty$ such that $|f(x_{n})-l| \geq \epsilon$. Now ...
-2
votes
0answers
25 views

Convergence of series (putnam training) [closed]

Does the series $\sum_{n=1}^{\infty} \frac {|\sin n|}{n} $ converge?
1
vote
1answer
43 views

Graphs with the property that $f=f^{-1}$

Ive been working on this question and can't seem to progress, I know that for a function to have the property $f=f'$ it must be symmetrical about the line $y=x$,I can't find reasoning behind in the ...
0
votes
0answers
30 views

How to convert vector field from cartesian to spherical

I have a vector field $A ( r) = \omega \times r$, where $r=(x,y,z)^T$ and now I want to express this field in cylindrical coordinates. How do I do this?
0
votes
2answers
25 views

Continuous Function on closed interval

I am having trouble understanding what this question is asking , by "$f$ has a zero" does it mean "there exists $x$ $\in$ $[a,b]$ such that $f(x)$=$0$? any help on how to answer this question in both ...
1
vote
0answers
32 views

Sequence of increasing compact sets

Suppose $X$ is a locally compact metric space which is $\sigma$-compact. Let $K$ be a compact subset of $X$. We can find a sequence of compact sets $K_{n}$ such that $K_{n} \subset \textrm{int}(K_{n + ...
0
votes
0answers
17 views

measurability restriction operator

Let $M\subset \mathbb{R}^k$ compact. For every $x\in M$ we define $L(x): \mathbb{R}^m \rightarrow \mathbb{R}^m $ a linear isomorphism Let $G_n (\mathbb{R}^m)=\{ W: W\ \mbox{is subspace of} \ ...