2
votes
1answer
18 views

What assumptions are needed to get two integrals close to each other?

I have functions $A,B,C$, where $\int_{\mathbb{R}} |A\cdot B - C| <\varepsilon$, and want to be able to say that $\int_{\mathbb{R}} A \approx \int_{\mathbb{R}} \frac{C}{B}$. What extra assumptions ...
0
votes
1answer
13 views

Close fourier transforms implies close time domain functions?

What conditions do we need so that $A(f)\approx B(f) \Rightarrow \mathcal{F}^{-1}\left\{A\right\}(t)\approx\mathcal{F}^{-1}\left\{B\right\}(t)$ The Fourier transforms of my two things look alike. In ...
0
votes
0answers
17 views

What is the contour lines of $\frac{2x+y}{2x-y}$?

What is the contour lines of $$f(x)=\frac{2x+y}{2x-y}$$? I need help to describe them... Id like to get help... Thank you!
0
votes
3answers
71 views

a continuous function, satisfying $f(α) = f(β) +f(α −β)$ for any $α, β ∈ \mathbb{R}$ [duplicate]

Hi need some help with this problem: Assume $f : \mathbb{R} → \mathbb{R}$ is a continuous function, satisfying $f(α) = f(β) +f(α −β)$ for any $α, β ∈ \mathbb{R}$, and $f(0) = 0$. Then $f(α) = α ...
0
votes
0answers
18 views

Mean value of a function over the n-sphere superficie.

We know that we can use the bloch sphere to represent an unitary vectors $v$ in $\mathbb{C}^{2}$, due to the fact $su(2) \approx so(3)$. Then, if we have the function $f:\mathbb{C}^{2} \rightarrow ...
1
vote
1answer
30 views

prove $\lim_{n\to\infty}\sup\{|f(x) - f_n(x) | : x \in S \} =0$

I understand how the theorem works but how would you prove that a sequence $f_n$ of functions on set $S \subset \mathbb{R}$ converges uniformly iff $$\lim_{n\to\infty} \sup\{|f(x) - f_n(x) | : x \in ...
0
votes
1answer
60 views

Is this construction already a contradiction

Let's say we have $\ell^p$ for $p>2$ and $\ell^2 \subsetneq \ell^p$. Let $U$ be a closed subspace of $\ell^2$. Is it possible that $U$ is isomorphic to $\ell^p$ as a Banach space?- I hardly think ...
2
votes
2answers
60 views

$C^\infty$ approximations of $f(r) = |r|^{m-1}r$

Consider $f(r) = |r|^{m-1}r$ where $m \geq 1$. Is it possible to find $C^\infty$ functions $f_n$, such that $f_n \to f$ uniformly on compact subsets of $\mathbb{R}$, $f_n' \to f'$ uniformly on ...
1
vote
0answers
46 views

Transfer vector space properties to dual space

I am curious about this here (Actually, I don't know if my assumptions are true or not) a) Let $X$ be a Banach space that is isomorphic to $Y$, then $X^*$ is also isomorphic to $Y^*$. I sketched a ...
2
votes
2answers
51 views

How to find maximum/minimum of $y=\frac{x(x^2-x+2)}{x^2-9}$?

How can I find minimum and maximum of $y=\frac{x(x^2-x+2)}{x^2-9}$? In other words, points, where $y'=0$. My current steps are: $$y=\frac{x(x^2-x+2)}{x^2-9}=x-1+\frac{11x-9}{x^2-9}$$ ...
0
votes
2answers
48 views

$l^r \subset l^p$ and is it even a subspace

It is true that for $r<p$ and $r,p \in [1,\infty)$ we have that $l^r \subset l^p$. Is it true that $l^r$ cannot be isomorphic to a subspace of $l^p$?
1
vote
1answer
35 views

Is $l^p$ closed in $L^p$?

Let's assume we have a subspace $X$ of $L^p$ and we know that $X \cong l^p$(this should just mean isomorph no isometry is assumed here). Can we infer from this that $X$ is closed? I have just read a ...
1
vote
1answer
42 views

Basic sequence- what is so special about it?

Let $(x_n)$ be a Schauder basis of a vector space $X$. This means that the $span(x_n)$ is dense in $X$, right? Then wikipedia introduces the notion of a $\textbf{basic sequence} $ when $(x_n)$ is a ...
1
vote
2answers
101 views
+200

Biorthogonal functions in $L^p$

I asked one question that is already answered: 1.) I have a question about Lemma 9.5 on page 93/94 reference. It's about the part of the proof where the sequence of $(g_n^*)$ are introduced. I don't ...
5
votes
1answer
77 views

Perturbation theory PDEs

I have the solution of a PDE of the form: $$ \Delta \Psi(r,\theta, \phi) = k \Psi(r,\theta,\phi)$$ on a set $\mathbb{R}^3 \backslash B(0,R)$. Hence, the actual solution is known there! Regarding ...
0
votes
2answers
58 views

old prelim exam question

In studying for my prelims, I can't quite come up with a solution to the following problem, from a few years ago. It has the feel of a "write the correct thing down, and it's 3 lines" problem. ...
3
votes
0answers
37 views

Riesz Fischer theorem?

I was wondering about the following: I read that Fischer-Riesz says that $L^2([0,1])$ is isomorphic to $l^2(\mathbb{N})$. Now it is obvious, that this should not depent on the fact which compact ...
0
votes
0answers
39 views

When integration by parts applies?

I think integration by parts is really useful, but I don't know when integration by parts applies. Take the following proposition as an example: Let $\Omega \subset \mathbb{R}^n$ be an open set, ...
3
votes
1answer
29 views

Limit of a hyperpower function

i have a question regarding this class of equations: Let $\gamma(x)=x^x$ Let $\Psi_n(x)=\underbrace{\gamma(x)\circ\gamma(x)\circ\gamma(x)}_n$, such that $\Psi_1(x)=\gamma(x)$ and ...
4
votes
1answer
102 views

A inequality of calculus [duplicate]

Let $f \in C^2[a,b]$ and $f(a) = f(b) = 0$, $f'(a) = 1$,$f'(b) = 0$, prove that $$\int_a^b|f''(x)|^2\,dx \geq \frac{4}{b-a}$$ Remark: This question is in the book functional analysis of Peking ...
5
votes
1answer
71 views

Spectral theorem in Quantum Mechanics

In Quantum Mechanics one often looks at self-adjoint(unbounded and closed) linear operators $A,B$ that are defined dense on $L^2$. My question is: Is it true that if we have $[A,B]=0$, where $[,]$ is ...
0
votes
1answer
22 views

question about the continuity of a function

I need to show that: If $f$ is continuous at $x_0$ iff $f^*(x_0)=f_*(x_0)$ where: $f^*(x_0)=\lim_{x \to x_0} \sup f(y)=\inf_{\epsilon > 0} \sup_{|y-x_0|<\epsilon}f(y)$ and $f_*(x_0)=\lim_{x ...
0
votes
0answers
27 views

Differentiating the sum ($\sum_{j=1}^\infty f_j(t)v_j(x,t)$) term by term

Let $f(x,t) = \sum_{j=1}^\infty f_j(t)v_j(x,t)$ where $f_j \in C^1(0,T)$ and $v_j(\cdot,t) \in C^\infty(\Omega)$ where $\Omega \subset \mathbb{R}^n$ is a domain. The sum makes sense in the sense that ...
2
votes
4answers
145 views

Concerning $C^0[0,1]$ and the $L^1$-Norm.

Consider the well known Euler sequence of functions $x^n$ ($n=1,2,3\ldots$) on $[0,1]$. It is clear that it converges against $\chi_{1}$, the characteristic function of the singleton $1$, in the ...
0
votes
0answers
20 views

What functions $f(t)$ satisfy $f(t) > \int_0^t e^{\xi-t} f(\xi)\,d\xi$?

Is there anything that can be said about a function $f(t)$ which satisfies $$f(t) > \int_0^t e^{\xi-t} f(\xi)\,d\xi$$ for some $t >0$? For example, if $f(t)$ satisfies this relationship, is it a ...
1
vote
1answer
44 views

$T_n \rightarrow T$ then we have $||T|| \le liminf(||T_n||)$

I know how to show that a cauchy sequence of linear continuous operators $T_n:X \rightarrow Y$ has a limit that is also such an operator(if Y is a Banach space), but I found this relation here too ...
0
votes
0answers
37 views

calculation of Simple terms

If $a>0$, $b\ne 0$ and $c \ge 0$ for which $\lambda$, $e^{\lambda x}[a\lambda^2-b\lambda-c]+ce^\lambda\ge1$ I really need this for some proof. $x\in[0,1]$ If I obtain $\lambda$ I can complete the ...
3
votes
2answers
77 views

Why the image of a linear map is not always a Banach space?

I have a question: Let's think about the map $T:V \rightarrow \text{ran}(T)$ and $V$ be a Banach space. Then we have that this is the same as the quotient map $[T]:V \rightarrow V/\ker(T)$ where the ...
10
votes
1answer
94 views

Non-divergence form of a 2nd order PDE

This might be a trivial question but I'm very rusty with regards to calculus and am new to PDEs. How would you write the following second order quasilinear equation in it's non-divergence form: The ...
1
vote
1answer
48 views

When can an arbitrary function be put into this form?

A problem I've been trying to address but not been able to get very far with is to devise a method to check whether or not some function $q(x)$ can be written like $$ q(x) = ...
2
votes
1answer
53 views

Orthogonal complement examples

I am looking for an example such that in a pre-Hilbert space $H$ we have for a subspace $U$ that (i) $\bar{U} \oplus U^\perp \neq H$ (ii) $ \bar{U} \neq U^{\perp \perp}$ Since finite and closed ...
2
votes
0answers
85 views

Fundamental theorem of calculus of Banach-space valued functions

Let $f:[a,b]\rightarrow E$ be a continuous function from the interval $[a,b]$ to a Banach space $E$. Let $F(x)=\int_a^xf(t)\text{ }dt$ where the integral is the Bochner integral. I have to prove that ...
6
votes
1answer
205 views

Continuous linear map

I am trying to understand how to start with this exercise. Let $\Omega \subset \mathbb{R}^n$, $n\ge 3$, open and bounded and $$ C^{1,b}(\Omega)= \{\,f\in C^1(\Omega): \text{$f$ and all its partial ...
3
votes
3answers
183 views

Is the sphere compact?

Riesz' lemma gives us that in infinite-dimensional spaces no ball is compact. but what is about the sphere$=\{x \in X; ||x||=1\}$? can we say something about the compactness of the sphere in ...
0
votes
1answer
59 views

Question about step in proof of Schauder's theorem

The statement is the following: Let $X,Y$ be Banach spaces and $T:X \rightarrow Y$ be a continuous linear operator. Then is $T'$ compact iff $T$ is compact. I have already understood the implication ...
2
votes
1answer
89 views

Euler Lagrange equation derivation and application of the fundamental lemma of the calculus of variations

Say we have: (1) $J(x) = \int_{\textit{to}}^{\textit{tf}} g(x(t),\dot{x}(t),t) dt$. We go through the general derivation and arrive at: (2) $\delta J(x,\delta x) = ...
0
votes
1answer
44 views

Minkowski functional and strange theorem

I have a theorem that says the following: Let X be a normed space and $U\subset X$ a convx subset with $0 \in \text{int(U)}$, then we have: $U$ is absorbing and if $\{x;||x|| < \epsilon\} \subset ...
2
votes
3answers
58 views

Are the convergent sequences dense in the bounded sequences?

Since it would be comfortable for something I am currently trying to prove if this would hold I wanted to ask here whether it is true that $c$ is dense in $l^{\infty}(\mathbb{N})$?
1
vote
1answer
91 views

Prove that the sequence of L-Lipschitz functions converge

$f_n(x): [a,b] \to \mathbb R$ are a sequence of functions that all are $L$-Lipschitz: means - $|f_n(x)-f_n(y)| \le L|x-y|$ , ($L$ is for all the functions) and assume $f_n \to f$ in a pointwise ...
1
vote
1answer
58 views

Minimizing a linear function on a strictly convex set.

All the theorems that I know considering the uniqueness of a solution to a minimization/maximization problem requires the strict convexity/strict concavity of the objective function. But consider the ...
0
votes
0answers
54 views

Equivalent differential equation problem to this fredholm integral equation.

I have heard that it is sometimes possible to transform Fredholm integral equations into differential equations. Could anybody here show me how to do this with the integral equation $$x(s)-\int_0^1 ...
1
vote
0answers
32 views

Differentiability of the Value (Support) Function

Consider the following problem, \begin{align} c(y,\mathbf{w})=\inf_{\substack{\mathbf{x} \in \mathbb{R}^n_{+} \\ \text{s.t. }f(\mathbf{x}) \geq y }} \mathbf{w} \cdot \mathbf{x} \end{align} where ...
0
votes
1answer
51 views

Prove range of f',$\{f'(x),x\in X\}$ dense in $X^*$

Let $X$ be a Banach Space and let $f: X\rightarrow \Bbb R$ be a Fre'chet differentiable function. Suppose that $f$ is bounded from below on any bounded set and satisfies $lim_{||x||\rightarrow ...
1
vote
1answer
41 views

Conditions on a sequence of functions to satisfy a certain simple! condition in limit

Let $f_n(x):[0,1]\to [0,1]$ be a sequence of continuous functions such that $f_n(x)\leq 1-x$ and $\int_0^1 f_n(x)\frac{1}{1-x} dx=\frac{1}{n}$. I am interested to know what extra conditions I must ...
2
votes
1answer
39 views

Prove lower bound of integral

I have a continuous function $h:[a,b]\rightarrow\Bbb C$. Let $$M=\sup_{x\in [a,b]}|h(x)|$$ I need to find function $f\in L^2[a,b]$ with: $${||f||}^2=\int_{a}^b|f(x)|^2dx=1$$ such that: $$ ...
0
votes
1answer
60 views

Prove x(t) is bounded given a integral inequality

I want to answer the following question: $x=x(t)$ is defined and continuous on $[0,T)$ and satisfies an integral inequality $$1 \leq x(t) \leq A_1 + A_2\int_0^t x(s)\big(1+\log x(s)\big) ds$$ for ...
0
votes
1answer
180 views

Quotient norm and actual norm

I have a question about the proof that $X\backslash U$ is a Banachspace if $X$ is one and $U$ is closed. In my book it is said, that for $x_k \in X$ and a series $\sum_{k=1}^{\infty}||[x_k]||< ...
0
votes
1answer
48 views

Does it matter in functional analysis whether we know (something about) a basis?

Let's look at a few spaces in functional analysis: $(L^p,C^n([0,1]), l^p,c,c_0,d)$ I actually only know the basis of one of these spaces. Which is the one that belongs to d, given by the unit ...
2
votes
2answers
269 views

Is there any handwavy argument that shows that $\int_{-\infty}^{\infty} e^{-ikx} dk = 2\pi \delta(x)$?

It should not be a good argument but rather a short one and one that convinces a physicist ( so no need for mathematical rigor ) that shows that $\int_{-\infty}^{\infty} e^{-ikx} dk = 2\pi \delta(x)$ ...
0
votes
1answer
87 views

Which of these norms are equivalent to the canonical one

Regarding the space of continuously differentiable functions $C^1([0,1])$, I am wondering which of these norms are equivalent to the norm $||x||= ||x||_{\infty} + ||x'||_{\infty}$. The candidates are ...