All Questions
Tagged with lp-spaces real-analysis
2,029
questions
4
votes
1
answer
81
views
weak convergence and pointwise implies $L_p$ convergence
Suppose $f_i \to f$ weakly in $L^p(X, M, \mu)$, $1 < p < \infty$, and that $f_i \to f$ pointwise $\mu$-a.e. Prove that $f_i^+ \to f^+$ and $f_i^- \to f^-$ weakly in $L^p$.
My proof:
Since $f^\pm ...
- 1,575
1
vote
3
answers
98
views
Show that the linear functional is unbounded in $C_{00}$. defined as $T$ is defined as $T(x)=\sum_{n=1}^{\infty} \frac{x_n}{\sqrt{n}},$
Given a linear functional $T: C_{00}\to C_{00}$. Where $C_{00}$ is space sequences with finitely many non-zero terms with $\ell_2$ norm.
$T$ is defined as $$T(x)=\sum_{n=1}^{\infty} \frac{x_n}{\sqrt{n}...
- 520
1
vote
1
answer
47
views
What justifies the use of global coordinates when computing the $L^p(\mathbb{T}^n)$ norm?
Consider the $n$ dimensional torus $\mathbb{T}^n$. The $L^p$ spaces over $\mathbb{T}^n$ is defined as consisting of an equivalence class of functions satisfying:
$$\int_{\mathbb{T}^n}|f|^p < \infty....
- 6,275
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
1
vote
2
answers
106
views
$L^{\infty}$ (uniform) decay of Dirichlet heat equation $u_t=\Delta u$
Let $\Omega$ be a smooth bounded open subset of $\mathbb{R}^N$. Consider the following initial-boundary value problem for the heat equation:
\begin{equation}
\begin{cases}
u_t=\Delta u\quad\quad\quad\;...
- 795
1
vote
2
answers
49
views
$L^\infty(\Omega)$ is dense in $L^{p,\infty}(\Omega)$ if $\Omega$ is compact
Given a compact set $\Omega\subset \mathbb{R}^N$, I am wondering if $L^\infty(\Omega)$ is dense in the weak $L^p$ space $L^{p,\infty}(\Omega)$ with $1< p<\infty$ (see here the definition).
I ...
- 433
3
votes
1
answer
167
views
A problem in L1 space
Problem: Let $(X, \mathcal{A}, \mu)$ be a measure space. Let $f: X \to [0, \infty)$ be measurable. Then define the set $$A_f = \left\{g \in L^1 (\mu)\ |\ |g| \leq f\mbox{ a.e.} \right\}.$$
Prove the ...
- 596
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
2
votes
2
answers
95
views
$| |x + y|^p - |x|^p | \leq \epsilon |x|^p + C |y|^p$
I want to demonstrate that: Let $1 < p < \infty$; for any $\epsilon > 0$, there exists $C = C(\epsilon) \geq 1$ such that for all $x, y \in \mathbb{R}$, we have
$$ | |x + y|^p - |x|^p | \leq \...
- 43
1
vote
0
answers
40
views
Decay at infinity of $L^2(\mathbb{R}^n)$ functions
I am trying to justify that a (normalized) solution $\phi$ in $L^2(\mathbb{R}^n)$ of:
$-\Delta\phi+f(x)\phi=K\phi$, with $f(x)=0$ in $\Omega$, $f(x)=M$ in $\Omega^c$
has to vanish outside $\Omega$ ...
- 21
0
votes
0
answers
44
views
Is $u(x)=\frac{1}{|x|^{\alpha}}$ in $W^{1,p}(B_1(0))$?
Consider a function $$ u(x)=\frac{1}{|x|^{\alpha}} \quad x\in B_1(0) \subset \mathbb{R}^N. $$
I should find condition about $p, N, \alpha$ for $u$ to be in $W^{1,p}(B_1(0))$. Following different books ...
- 133
2
votes
1
answer
37
views
Proof: If $\mu$ is $\sigma$-finite and $\mathscr{A}$ is countably generated, then $L^p(X,\mathscr{A},\mu)$ $1\leq p<+\infty$ is separable.
Background
I have some trouble understanding a step of the proof of the following proposition:
Proposition$\quad$ Let $X$ be a measure space, and let $p$ satisfy $1\leq p<+\infty$. If $\mu$ is $\...
- 2,493
2
votes
1
answer
62
views
For $1\le p < +\infty$ $L^p$ is a Banach space: Real and abstract analysis, Hewitt - Stromberg
I have some doubts about the proof of this theorem. From time to time I will put my justification.
For $1\le p < +\infty$, $L^p$ is a Banach space
Let $(f_n)_n$ be a Cauchy sequence in $L^p$, i.e., ...
- 47
2
votes
1
answer
38
views
Proving that convergence of norms and convergence a.e. implies strong convergence
I have in my notes the following theorem
Theorem
$(Y,\mathcal{F},\mu)$ $\sigma-$finite measure space, $p\geqslant 1$, $\{f_n\}\subset L^p(Y)$ sequence of functions, $f\in L^p(Y)$ such that $$\lim_{n\...
- 516