Questions tagged [lp-spaces]
For questions about $L^p$ spaces. That is, given a measure space $(X,\mathcal F,\mu)$, the vector space of equivalence classes of measurable functions such that $|f|^p$ is $\mu$-integrable. Questions can be about properties of functions in these spaces, or when the ambient space in a problem is an $L^p$ space.
5,632
questions
3
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1
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222
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For $0<p<1$ showing $\Big(\int_\Omega |f|^pd\mu\Big)^{1/p}\leq \frac{1}{R_0} + \Big(\int_\Omega |f_{R_0}|^p\Big)^{1/p} $
In this problem Limit of $L^p$ norm when $p\to0$, the writer states that for $0<p<1$ we have that
$$\Big(\int_\Omega |f|^pd\mu\Big)^{1/p}\leq \frac{1}{R_0} + \Big(\int_\Omega |f_{R_0}|^p\Big)^{1/...
3
votes
1
answer
73
views
Showing weak convergence of a sequence in $L^p(R)$
I have a sequence of functions $f_k$ in $L_p(R)$, with $1<p<\infty$ and I'd like to show that it weakly converges to $0$.
This is the sequence, where $k\in N$
$$f_k = 1_{[k,k+1]}$$
What I've ...
0
votes
1
answer
46
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Convergence in $L^2(]0,T[,v)$
if I have a sequence $(u_n)_n$ that converges to $u$ in $L^2(]0,T[,v)$ , then why $ ||u_n(.)||^2_v \rightarrow ||u(.)||^2_v$ in $L^1(]0,T[)$?
Remark that $L^2(]0,T[,v)$ is the space of measurable ...
1
vote
1
answer
56
views
Show that $\frac{1}{x^{1/p}(\ln(2/x))^{2/p}}$ is only in $L^p(0,1)$
Problem: Let $\mu$ be the Lebesgue measure.
Define $f:(0, 1) \to \mathbb{R}$,
$$
f(x) = \frac{1}{x^{1/p}(\ln(2/x))^{2/p}}.
$$
Show that $f \in L^p(0, 1)$. Show for every $q>p$ that $f \not\in L^q(...
2
votes
1
answer
115
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(Understanding the proof of) Riesz representation theorem in $L^p$
I'm studying $L^p$ spaces for the first time. I am following, among others, the book by Stein & Shakarchi Functional Analysis. Introduction to Further Topics in Analysis. Before the proof of the ...
4
votes
0
answers
63
views
Random signs inequality
Let $X_1, X_2, \dots, X_n, \dots$ be Rademacher random variables (random signs), i.e. with the distribution
$$X_i \sim \left\{
\begin{array}{@{}ll@{}}
\ \ \ 1 & \text{with probability} \ \ \...
0
votes
0
answers
20
views
A question about sequence convergence in $L^p(\mathbb{R}^n)$
If $u_m\rightarrow u$ in $H^1(\mathbb{R}^n)\times H^1(\mathbb{R}^n)$(weak) and $u_m\rightarrow u$ in $L^4(\mathbb{R}^n)\times L^4(\mathbb{R}^n)$, denote $u_m=(u_{m,1},u_{m,2})$ and $u=(u_{1},u_{2})$, ...
0
votes
1
answer
50
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Estimating $\int_{\mathbb{R}}|\frac{\log|x|}{1+x^4}|\ dx$
I'd like to show that $\frac{\log|x|}{1+x^4} \in L^1(\mathbb{R})$, but I have trouble with estimating
$$\int_{\mathbb{R}}\left|\frac{\log|x|}{1+x^4}\right|\ dx.$$
Writing $(1+x^4) = (x^2 - i)(x^2 + i)$...
-1
votes
1
answer
45
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Uniform convergence of sum to constant, plus $L^2$ convergence of summands to constant, implies uniform convergence of summands? [closed]
If one has $f_N,g_N \geq 0$ continuous, defined over some compact sets (assumed of measure one for simplicity), such that:
$||f_N+g_N||_{L^1} = 1$;
$f_N + g_N \to C$ in $L^\infty$;
$f_N \to C_f$ in $...
2
votes
1
answer
1k
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Prove that $L^{\infty} [0,1]$ is not separable.
Prove that $L^{\infty} [0,1]$ is not separable.
I know that $\ell^{\infty} = L^{\infty} (\Bbb N, \mathcal P (\Bbb N), \mu_c)$ is not separable where $\mu_c$ is the counting measure on $\mathcal P (\...
3
votes
1
answer
212
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$T(x_1,x_2,x_3,x_4,\ldots)=\left(x_2,\frac{x_3}{2},\frac{x_4}{3},\frac{x_5}{4},\ldots\right)$, spectral radius and spectrum
Define a pair of operators $S\colon\ell^2 \rightarrow \ell^2$ and $L\colon\ell^2 \rightarrow \ell^2$ (of sequences of complex or real numbers) as follows:
$S(x_1,x_2,x_3,x_4,\ldots)=\left(x_1,\frac{...
1
vote
2
answers
84
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Can I say that $\int f_n\ d\mu \to \int f\ d\mu$?
Let $(X,\mathcal F, \mu)$ be a finite measure space. Let $\{f_n\}_{n \geq 1}$ be a sequence of $\mathcal F$-measurable functions converges to a $\mathcal F$-measurable function $f.$ Let $1 \lt p \lt \...
1
vote
0
answers
45
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Lp Spaces theorems
In the following theorems:
Theorem. Let $(X,\mathcal{F},\mu)$ be a measure space and $1\le p\le\infty$. For each $f\in L^p(\mu)$ and $\epsilon>0$ there exists a simple function $f_s$ such that $|...
1
vote
1
answer
99
views
Using Cavalieri to derive a formula for $n$-dim $L^p$ ball
let $p>0$ and $B_p^n:=\{x\in \mathbb{R}^n : |x_1|^p+\cdots + |x_n|^p\leq 1\}$. Show that $B_p^n$ is Borel-measurable with $$\lambda^n(B_p^n)=\frac{(2\Gamma(1+\frac{1}{p}))^n}{\Gamma(1+\frac{n}{p})}$...
1
vote
0
answers
39
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Showing $\|u\|_{L^{\infty}}\le C\|u\|_{W^{1,p}}$.
I was reading the proof from Brezis' book that show that if $p>N$ we have $W^{1,p}(\mathbb{R}^N)\subset L^{\infty}(\mathbb{R}^N)$. He starts proving $\|u\|_{L^{\infty}}\le C\|u\|_{W^{1,p}}$ for ...
1
vote
0
answers
29
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study the convergence of this sequence of functions in $L^{p}(\mathbb{R}^{d})$
Take $\alpha >0$ and for all $x\in\mathbb{R}^{d}$ define the sequence:
$$u_{n}(x)=\min\{1,|x|^{-\alpha}\}\chi_{B(0,n)}(x)$$
where $\chi$ denotes the characteristic function. I have to study strong ...
4
votes
1
answer
76
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Singularity at 0 is at most a pole of order 7 if the complex function is 1/4 integrable.
So the real question is, given f analytic in the punctured disk $(D_1 \backslash \{ 0\})$ and $\int_{D_1} |f(z)|^{1/4} < \infty$, characterize the smoothness of f at 0.
It is easy to show that if $...
0
votes
1
answer
252
views
$f_n\xrightarrow{L^p}f \implies f_n'\xrightarrow{L^p}f'$?
Let $(f_n)_{n\ge 1}$ be a sequence of function that is differentiable. such that $f_n\xrightarrow{L^p}f$ converges. I was wondering if $f_n'\xrightarrow{L^p}f'$. So far I could only find this result ...
3
votes
1
answer
172
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Proof that a particular subset of $L^2[\Pi]$ is dense and first category set (Baire's category)
I hope I don't make too many mistakes since this is my first post.
I was trying to prove that $$M = \left[ f\in L^2[\Pi] : \sum_{n=-\infty}^{+\infty}\hat f(n) \quad converges \right]$$ is dense and ...
1
vote
1
answer
318
views
How to prove the lower semi-continuity of this functionnal on $H^1(\Omega)$
I am studying the following application which goes from $H^1(\Omega)$ to $\mathbb{R}$ with $\Omega$ a bounded regular subset of $\mathbb{R}^3$ :
$$H : u \mapsto \int_{\Omega} \left(|\nabla u|^2 - a \...
2
votes
1
answer
445
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Relations between Bochner spaces $L^p(0,T;L^q)$
Let $(X,\|\cdot\|_X)$ be a Banach space. The function $u=u(t,x)$ belongs to Bochner space $L^p(0,T;X)$ if the norm
$$ \|u\|_{L^p(0,T;X)} = \left(\int_0^T \|u(t,\cdot)\|_X^p \mathrm{d}t\right)^{1/p} \...
1
vote
1
answer
314
views
Weak* convergence in $L^\infty$ and the strong convergence in $L^2$ of a mollification?
I feel like this should be obvious to me but I'm blanking.
Let $\Omega$ be an open bounded subset of $\mathbb R^n$. Let $f_n\in C^\infty_c(\Omega)$ and $f\in L^\infty(\Omega)$ be such that
$$ f_n\...
-1
votes
1
answer
348
views
Can the limit of a sequence of functions in $L_p$ and the limit in almost everywhere convergence be different? [duplicate]
I'm sure this is an easy question, but I am somewhat confused. I'm considering here a measure space $(X, \mathcal{A}, \mu)$, and the functions are real valued.
Given a sequence of functions $(f_n) \in ...
1
vote
0
answers
294
views
Q: Exercise 6.R Bartle
I'm trying to solve exercise 6R from Bartle's "Elements of Integration" that states:
Let $(f_n)$ be a Cauchy sequence in $L_p$. Then there exists a measurable set $E_\epsilon$ with $\mu(E_\...
0
votes
1
answer
24
views
How to prove this function is $L^{r}(E)$
Let $p,q\in[1,\infty)$, $E\subseteq\mathbb{R}$ measurable with finite measure. Let $u\in L^{p}(E)\cap L^{q}(E)$, fix $\alpha\in[0,1]$ and define:
$$\frac{1}{r}=\frac{1-\alpha}{p}+\frac{\alpha}{q}$$
...
0
votes
1
answer
127
views
How can we show $\left\|g\ast f\right\|_{L^p}\le\left\|g\right\|_{L^1}\left\|f\right\|_{L^p}$?
Let $p\ge1$, $f\in L^p(\mathbb R^d)$ and $g\in L^1(\mathbb R^d)$. I've read that the inequality $$\left\|g\ast f\right\|_{L^p}\le\left\|g\right\|_{L^1}\left\|f\right\|_{L^p}\tag1$$ would follow from ...
0
votes
1
answer
44
views
Study the convergence of this sequence in $L^{2}(\mathbb{R})$
Let $x_{n}$ be a seuquence in $\mathbb{R}$, define the sequence of functions:
$$u_{n}=\chi_{B(x_{n},1)}$$
where $\chi$ denotes the characteristic function. Study the convergence of $u_{n}\in L^{2}(\...
0
votes
0
answers
111
views
Examples of $L^p \cap L^1$ convergence
I'm struggling to find examples of a sequence of functions in two different cases: let $1<p<2$
functions $(f)_n \in L^p(\Bbb{R},dx)\cap L^1(\Bbb{R},dx)$ such that $(f)_n$ converges strongly in $...
1
vote
1
answer
85
views
Continuity in the mean: $\lim_{y \to 0}\int_{\mathbb{R}^d}|f(x+y)-f(x)|dx=0$
Let $f \in L_{\mathbb{R}^+}^1(\mathbb{R}^d,\mathcal{B}(\mathbb{R}^d),\lambda_d).$ Consider the sequence $f_k=\min(f,k)\min(1,\max(0;k+1-|x|))$ to show the continuity in the mean:
$$\lim_{y \to 0}\int_{...
1
vote
1
answer
54
views
Seemingly simple inequality between $L_p$ norms (When is $\|f\|_p \leq \|f\|_2$?)
I've found the following statement on a book:
It is easy to see that given any $p < 2$, if a function $h$ on a probability space has very small support, then its $L_p$ norm is much smaller than ...
1
vote
1
answer
80
views
$L^0( \mathbb P | \mathcal F_0 ; || \cdot ||_{\mathbb R^d} ) \subset L^0( \mathbb P | \mathcal F_1 ; || \cdot ||_{\mathbb R^d} ) $?
As stated in the title,
How to prove that $$L^0( \mathbb P | \mathcal F_0 ; || \cdot ||_{\mathbb R^d} ) \subset L^0( \mathbb P | \mathcal F_1 ; || \cdot ||_{\mathbb R^d} ) $$
where the space $ L^0( \...
1
vote
1
answer
591
views
Is every bounded function in $L^p$ for any $p \ge 1$?
Disclaimer: This might be very trivial, but I haven't learned any of this formally. (I didn't know what a Lebesgue integral was before looking it up today.) It is possible I am not understanding the ...
-1
votes
2
answers
66
views
Counter-example to the extended dominated convergence theorem
To add some context, let $(\Omega,\mathcal{F},\mu)$ be a measure space. The statement of the Extended Dominated Convergence Theorem(EDCT) is:
$\lbrace f_{n}\rbrace_{n\geq 1}$ be a sequence of ...
3
votes
1
answer
254
views
Spectrum and compactness of $L^1$ operator
Consider a continuous complex function $g$ defined on $[0,1]$. On $L^1([0,1])$ consider operator
$$Tf(x) = g(x)f(x).$$
Calculate the norm. Determine its point spectrum and spectrum. Decide for what ...
1
vote
0
answers
44
views
$C([0,1], \ell_p)$ has a copy of $\ell_q$?
By Banach-Mazur theorem [ every separable Banach space X is isomorphic to a subspace of $C(0,1)$ ] we have that $C([0,1])$ has a copy of $\ell_p$.
But, $C([0,1], \ell_p)$ has a copy of $\ell_q$, where ...
1
vote
1
answer
69
views
Change of measure s.t. stochastic process is in $L^p$
Assume a filtered probability space $(\Omega, \mathcal{G},(\mathcal{G}_t)_{t \in [0,1]}, \mathbb{P})$ and $(S_t)_{t \in [0,1]}$ an adapted stochastic process (no further path properties).
Fix $1 \leq ...
3
votes
1
answer
124
views
Induction of topological space by an $L^2$-space
Let $(L^2(\Xi,\boldsymbol{\Xi},\mu;\mathbb{R}),\langle\cdot,\cdot\rangle)$ be a Hilbert space, where $L^2(\Xi,\boldsymbol{\Xi},\mu;\mathbb{R})$ denotes the set of all (equivalence classes of) $\...
1
vote
1
answer
37
views
How to prove uniqueness of approximate "jump" values of an $L^1$ function?
Let $v \in \mathbb{S}^{n-1}$ be a unit vector. Given $r>0$, Let $B_r(x)$ be the ball of radius $r$ around $x \in \mathbb{R}^n$, and define
$$B_r^+(x,v)=\{ y \in B_r(x) \, | \, \langle y-x,v\rangle &...
2
votes
1
answer
200
views
Interesting question about functionals in measure theory, any suggestion please.
Let $(X,A,\mu)$ be a measurable space. $1<p,q<\infty$, $\frac{1}{p}+\frac{1}{q}=1$.
$Z\subset L^p$ a subspace of $L^p$ .Let $\phi: Z\to C$ a linear functional.
Assume that there's $m<\infty$ ...
2
votes
2
answers
213
views
Spectrum and compactness of $\ell ^1$ operator
Consider complex sequence $(a_n)$, $|a_n| \leq 1$. On $\ell ^1$ space consider the operator
$$ T(x_n) = (a_n x_n).$$
Find the norm. Decide for what sequences is $T$ compact. Find point spectrum and ...
0
votes
1
answer
50
views
If $(f_n)_{n\in\mathbb N}$ converges in $L^p$, does the sequence of normalized functions converge as well?
Let $(E,\mathcal E,\mu)$ be a measure space and $p\ge1$. If $(f_n)_{n\in\mathbb N}\subseteq L^p(\mu)$ and $f\in L^p(\mu)$ with $$\left\|f_n-f\right\|_{L^p(\mu)}\xrightarrow{n\to\infty}0\tag1,$$ can we ...
0
votes
0
answers
46
views
Is $L^p(X^2) \cong L^p(X) \otimes L^p(X)$ true?
I was wondering if for $X$ a compact set of $\mathbb{R}^d$ with $d \in \mathbb{N}$: $p \in [1,+\infty]$ (especially for the case $p=2$ !) one have
$$L^p(X^2) \cong L^p(X) \otimes L^p(X)$$
0
votes
1
answer
267
views
Show that for each $g \in L^q([0,1])$, $\lim_{n \to \infty} \int_0^1 f_ng = \int_0^1 fg$.
Let $1 < p < \infty$. Let $\{f_n\}$ be a sequence of functions in $L^p([0,1])$ that converges almost everywhere to a function $f \in L^p([0,1])$. Also, suppose there exists a constant $M$ such ...
1
vote
1
answer
44
views
Disproving $ L^2(X, U(\mathbb{C},n)) = U(L^2(X,\mathbb{C}^n))$.
Here $X$ is some measurable space and $U(L^2(X,\mathbb{C}^n))$ denotes the Banach space of unitary automorphisms of the Hilbert space $L^2(X,\mathbb{C}^n)$.
Let $U(\mathbb{C},n)$ denote the unitary ...
1
vote
1
answer
38
views
The integral $\int_{U}u^{4}$ is well-defined for $u\in H_{0}^{2}(U)$ and $U\subset\mathbb{R}^{n}$ open, bounded, $n\leq3$.
I'm trying to prove that the integral $\int_{U}u^{4}$ is well-defined for $u\in H_{0}^{2}(U)$ and $U\subset\mathbb{R}^{n}$ open, bounded, $n\leq3$. In other words, I need to prove that $H_{0}^{2}(U)$ (...
1
vote
0
answers
24
views
In which space is this sequence converging?
I am reading proof on the partial regularity of harmonic maps and there is a part where the author considers a sequence with the following terms on the unit ball $B$ in $\mathbb{R}^2.$
$P_{\delta}\in ...
1
vote
2
answers
572
views
Does convergence in the $r$th mean for all $r$ imply almost sure convergence?
Almost sure convergence of a sequence of random variables and convergence in the $r$th mean (for any $r$) each imply convergence in probability. However, (I believe) neither of them implies the other ...
1
vote
0
answers
78
views
Homeomorphism with Hilbert Cube
Let $\{e_k\}_k$ be a basis for $ L^2(X,\mu)$, and $\{a_k\}_k$ limited sequence of real numbers.
$X$ is a compact metric space, and $\mu$ is the borelian measure.
Can one say the set $\{g \in L^2(X,\mu)...
1
vote
1
answer
216
views
Dual spaces, Am I doing right?
Let $\mu$ be the counting measure on the measurable space $(N,P(N))$.
$1 <p,q< \infty$ , $\frac{1}{p}+\frac{1}{q}=1$.
Define: $\phi: l_q\to (l_p)^*$ such as for all $(a_n)\in l_p$ and $(b_n)\in ...
1
vote
1
answer
2k
views
Prove that dual space of $\ell^1$ is $\ell^{\infty}$
Prove that dual space of $\ell^1$ is $\ell^{\infty}$
My attempt : I got the answer Here but im not able to understand the answer
we know that the norm of $ x\in \ell^1$ is given by $||x||_1=\sum_{k=...