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.

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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/...
Andrew Shedlock's user avatar
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 ...
Shravan Vinod's user avatar
0 votes
1 answer
46 views

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 ...
sara's user avatar
  • 305
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(...
hirohe's user avatar
  • 502
2 votes
1 answer
115 views

(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 ...
Juan Castaño's user avatar
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} \ \ \...
user avatar
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 $u=(u_{1},u_{2})$, ...
Kimura Leo's user avatar
0 votes
1 answer
50 views

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 $(1+x^4) = (x^2 - i)(x^2 + i)$...
Vicky's user avatar
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-1 votes
1 answer
45 views

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 $...
Andrea Fuzzi's user avatar
2 votes
1 answer
1k views

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 (\...
Anil Bagchi.'s user avatar
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3 votes
1 answer
212 views

$T(x_1,x_2,x_3,x_4,\ldots)=\left(x_2,\frac{x_3}{2},\frac{x_4}{3},\frac{x_5}{4},\ldots\right)$, 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{...
user1234567890's user avatar
1 vote
2 answers
84 views

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 \...
Anacardium's user avatar
  • 2,347
1 vote
0 answers
45 views

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 $|...
masacuota's user avatar
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})}$...
Analysis's user avatar
  • 2,450
1 vote
0 answers
39 views

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 ...
edamondo's user avatar
  • 1,277
1 vote
0 answers
29 views

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 ...
Davide Trono's user avatar
  • 1,431
4 votes
1 answer
76 views

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 $...
user3646987's user avatar
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 ...
edamondo's user avatar
  • 1,277
3 votes
1 answer
172 views

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 ...
Don Abbondio's user avatar
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 \...
Velobos's user avatar
  • 2,190
2 votes
1 answer
445 views

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} \...
M_S's user avatar
  • 652
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\...
Calvin Khor's user avatar
  • 34.9k
-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 ...
user480840's user avatar
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_\...
Fernando Nazario's user avatar
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: $$\frac{1}{r}=\frac{1-\alpha}{p}+\frac{\alpha}{q}$$ ...
Davide Trono's user avatar
  • 1,431
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 ...
0xbadf00d's user avatar
  • 13.4k
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}(\...
Davide Trono's user avatar
  • 1,431
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 $...
Lupetto1927's user avatar
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_{...
john's user avatar
  • 155
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 ...
Matheus barros castro's user avatar
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( \...
Marine Galantin's user avatar
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 ...
kanso37's user avatar
  • 358
-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 ...
Oogway's user avatar
  • 145
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 ...
user avatar
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 ...
ZHN's user avatar
  • 1,419
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 ...
cruiser0223's user avatar
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) $\...
Frederick Leuchte's user avatar
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 &...
Asaf Shachar's user avatar
  • 25.1k
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 $m<\infty$ ...
Lam18373's user avatar
  • 599
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 ...
user avatar
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 ...
0xbadf00d's user avatar
  • 13.4k
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 $$L^p(X^2) \cong L^p(X) \otimes L^p(X)$$
Bast's user avatar
  • 21
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 ...
TheSenate's user avatar
  • 564
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 ...
Overflowian's user avatar
  • 5,699
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 $H_{0}^{2}(U)$ (...
Calculix's user avatar
  • 3,354
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 ...
Student's user avatar
  • 9,196
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 ...
tparker's user avatar
  • 6,209
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)...
Mateus R.'s user avatar
  • 119
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 ...
Mat999's user avatar
  • 537
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=...
jasmine's user avatar
  • 14.5k

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