All Questions
415
questions
1
vote
2
answers
119
views
Prove that $T$ is not a compact operator.
Let $T:\ell_{2}(\mathbb{Z})\rightarrow \ell_{2}(\mathbb{Z})$ be the operator defined by,
$$T((x_i)_{i\in \mathbb{Z}})=((y_i)_{i\in \mathbb{Z}}).$$
where
$$
y_{j}=\frac{x_{j}+x_{-j}}{2}, \quad j \in \...
- 520
2
votes
0
answers
32
views
Linear Analysis – Examples 1-Q5 General $l^p$ spaces vs $l^1, l^\infty$
Let $1<p<\infty$, and let $x$ and $y$ be vectors in $l_p$ with $\left \|x \right \|=\left \|y \right \|=1$ and $\left \|x +y \right \|=2$, how to prove $x=y$?
I know how to prove for $p=2$ ...
- 451
0
votes
0
answers
27
views
Auxiliar inequality for Rellich-Kondrachov theorem
To prove the Rellich-Kondrachov Theorem it is used the following statement
If $u\in W^{1,1}(\Omega)$, with $\Omega \subset \mathbb{R}^N$ open, bounded and s.t. $\partial \Omega$ is $C^1$, then $||\...
- 133
0
votes
1
answer
45
views
Dominated convergence theorem for $L^{\infty}$ with additionnal hypothesis of vanishing at infinity
Let $f\in L^{\infty}(\mathbb{R}^n, \mathbb{R})$. Denote $\chi_R$ the characteristic function on $B(0,R)\subset\mathbb{R}^n.$ If $\underset{\|x\|\to \infty}{\text{lim}} f(x) = 0$, then will
$\underset{...
- 57
1
vote
1
answer
56
views
Problem on density of a subset on $L^2([a,b],\mathbb{R})$: looking for a better solution
I have this problem that the professor gave us:
Let $\gamma ,a,b\in\mathbb{R}$ and $$D_{\gamma}=\{u\in C^2([a,b],\mathbb{R}):\gamma u(a)-u'(a)=0,\gamma u(b)-u'(b)=0\}$$
Prove that $D_\gamma$ is dense ...
- 94
0
votes
0
answers
44
views
Is $u(x)=\frac{1}{|x|^{\alpha}}$ in $W^{1,p}(B_1(0))$?
Consider a function $$ u(x)=\frac{1}{|x|^{\alpha}} \quad x\in B_1(0) \subset \mathbb{R}^N. $$
I should find condition about $p, N, \alpha$ for $u$ to be in $W^{1,p}(B_1(0))$. Following different books ...
- 133
2
votes
1
answer
37
views
Proof: If $\mu$ is $\sigma$-finite and $\mathscr{A}$ is countably generated, then $L^p(X,\mathscr{A},\mu)$ $1\leq p<+\infty$ is separable.
Background
I have some trouble understanding a step of the proof of the following proposition:
Proposition$\quad$ Let $X$ be a measure space, and let $p$ satisfy $1\leq p<+\infty$. If $\mu$ is $\...
- 2,493
3
votes
1
answer
67
views
Prove that $\int _Xf_ngd\mu \overset{n\to\infty}{\to}\int _Xfgd\mu$ ,$\forall g\in \mathcal{L}^\infty (\mu )$ if it's true $\forall g\in C_b(X)$
Let $X$ be a Polish space and $\mu :\mathfrak{B}_X\to\overline{\mathbb{R}}$ a finite measure on the Borel subsets of $X$. Suppose $(f_n)_{n\in\mathbb{N}}$ is a sequence of $\mathcal{L}^1(\mu )$ and $f\...
- 1,209
6
votes
2
answers
96
views
Understanding the proof of $L^p(X,\mathscr{A},\mu)$ is complete ($1\leq p<+\infty$)
Background
I have some questions when reading the proof of $L^p(X,\mathscr{A},\mu)$ is complete for $1\leq p<+\infty$. The proof is proceeded by showing that each absolutely convergent series in $L^...
- 2,493
4
votes
2
answers
141
views
Understanding the proof of $L^{\infty}$ is complete.
I got lost when reading the proof of $L^{\infty}$ is complete. The book proceed the proof as follows:
We show that each absolutely convergent series in $L^{\infty}(X,\mathscr{A},\mu)$ is convergent. ...
- 2,493
1
vote
0
answers
58
views
Proof of Du Bois-Reymond Lemma using Riesz representation theorem
I’m working with this version of the fundamental calculus of variations lemma:
If $f\in L^p(\mathbb{R}^n)$ and $\int f\phi dx = 0 $ for all $\phi \in C^\infty_c(\mathbb{R}^n)$, then $f=0$ a.e.
My ...
- 133
1
vote
0
answers
49
views
Is every continuous function $g$ of the form $g(p) = \|f\|_p$ for some $f : \mathbb R \to \mathbb R$?
Let $1 \leq a \leq b \leq \infty$, and let $f \in L_a(\mathbb R) \cap L_b(\mathbb R)$. Then the function $g : [a,b] \to [0,\infty)$, $p \mapsto \|f\|_p$, is continuous. We also have that if there is $...
- 6,033
0
votes
0
answers
37
views
Translation in $L^{\infty}$
Consider the translation operator $\tau_h$ defined on $L^\infty(\mathbb{R}^n)$ s.t. $\tau_hu(x)=u(x-h)$. I know that $\tau_h$ is not continuous with respect to $h$, I mean it’s not true that $h\to 0$ ...
- 133
1
vote
0
answers
31
views
Boundedness of an Integral Operator on $L^p(\mu)$
I realize this question has been asked way too many times, for instance, it's this exact problem which I will put below for convivence:
Let $(X,\Omega, \mu)$ be a $\sigma$-finite measure space with $...
- 1,399
2
votes
0
answers
52
views
If $u\in L^1_{\text{loc}}(\Omega)$ and $\Delta u\in L^1_{\text{loc}}(\Omega)$, then $\nabla u \in L^1_{\text{loc}}(\Omega)$?
(And as well, a transcription for those unable to load images:)
Remark 2. There is a local form of Corollary 1, namely if $u\in L^1_{\text{loc}}(\Omega)$ and $\Delta u \in L^1_{\text{loc}}(\Omega)$, ...
- 525