For questions about $L^p$ spaces, that is, given a measure space $(X,\mathcal F,\mu)$, the vector space of equivalence class of measurable functions with $p$-th power of the absolute value integrable. Question can be about properties of elements of these spaces, or when the ambient space on a ...

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2
votes
1answer
29 views

For $(l^2,\|\cdot\|_2)$ and $e_n=(0,0,.,1,0,.)$ and a bounded linear functional $\Phi$ find $p\geq 1$ where $\sum_{n=1}^\infty |b_n|^p$ converges?

For $(l^2,\|\cdot\|_2)$ and $e_n=(0,0,...,1,0,...)$ and a bounded linear functional $\Phi$ find a value of $p\geq 1$ where $\sum_{n=1}^\infty |b_n|^p$ converges for $b_n=\Phi(e_n)$? Ok so since ...
0
votes
0answers
24 views

The dual of the space $L^\infty$. [duplicate]

As we know the dual of $L^p$s are $L^q$s where $\frac{1}{p} + \frac{1}{q} =1$, and dual of $L^1$ is $L^\infty$. What is dual of the space $L^\infty (E)$ where E is a measurable subset of $\mathbb{R} ^ ...
1
vote
0answers
30 views

Approximation of Conditional Expectation with Respect to “Y” Using Simple Approximation of “Y”

Background. (TL:DR you can skip to Question. below.) This is a followup question to one of my previous questions (linked here) on this website. In short, the other question was about how to express ...
2
votes
1answer
32 views

Proposed proof for convergence in Sobolev space

Consider the Anisotropic Sobolev Space defined by: $$W^{1,\overrightarrow{p},\epsilon}(\Omega) := \{ u \in L^{1+\frac{1}{\epsilon}}(\Omega), \frac{\partial u}{\partial x_{i}} \in L^{p_{i}}(\Omega), ...
1
vote
0answers
22 views

Prove that a function lies in $L^1$ and in $W^{(1,1)}$ for some parameter

I want to do the following tasks Let $G:=B_1(0)\subset \mathbb R^2$ be the open ball around $0$ with radius 2 in the norm $||\cdot||$ and $u_{\rho}(x)=||x||^{\rho}_2$, $x\in G$. Show the following ...
1
vote
1answer
37 views

If $f_n \to f$ and $g_n \to g$ in $L^p$, and $g_n$ is uniformly bounded, then $f_ng_n \to fg$

Problem: If $f_n \to f$ and $g_n \to g$ in $L^p$, and $g_n$ is uniformly bounded, then $f_ng_n \to fg$ in $L^p$. An official solution I saw for this problem looked very different. Here is my ...
3
votes
1answer
29 views

Relation between the modulus of integrability and $L^p$ spaces

Let $(X,\mu)$ be a measure space with $\mu(X)<\infty$. Given an integrable function $f$ on $X$, we can quantify its integrability in multiple ways. One is the modulus of integrability, which is a ...
0
votes
1answer
40 views

Does the sequence $( n^{1/n} -1)$ belong to any $\ell^p$ space? [on hold]

The sequence $( n^{1/n} -1)$ converges to zero but does this sequence belong to the $\ell^p$ space for $p\in\mathbb R$? I don't know the answer, or how one would prove it. Same problem with the ...
3
votes
1answer
40 views

If a sequence $f_n$ is bounded in $L^2$ and converges to zero a.e., then $f_n\to 0$ in $L^p$ for $0<p<2$

Let $M>0$, $\{f_n\}\subset L^2([0,1])$ such that $\int_0^1 |f_n|^2 dm\leq M$ and $f_n(x)\to 0$ as $n\to\infty$ almost everywhere, $m$ is Lebesgue measure. Show that for all $0<p<2$, ...
2
votes
1answer
32 views

Weak Convergence for a specific example

Let $X=(0,1)$ and $u_v(x) = v^{1/p}1_{(0,1/v)}$, where $1 < p < \infty$. It needs to be shown that $\lim_{v \to \infty} \int_X u_v \phi = 0$ for all $\phi \in \mathcal{L}^{p^\prime}$ (The ...
2
votes
1answer
31 views

The derivative of a $L^ {\infty}$ function

If I take the derivative of a function in $L^ {\infty}$ (that is, the function is bounded by a number) in any direction, in which space the derivative is defined? Are there some properties for ...
4
votes
1answer
31 views

Partial Converse of Holder's Theorem

Holder's Theorem is the following: Let $E\subset \mathbb{R}$ be a measurable set. Suppose $p\ge 1$ and let $q$ be the Holder conjugate of $p$ - that is, $q=\frac{p}{p-1}.$ If $f\in L^p(E)$ and $g\in ...
1
vote
1answer
34 views

Extreme point of unit balls, over $\mathbb C$

I've been trying to determine what are the extreme point of the unit balls of $\ell^1$ and $\mathcal{C}[0,1]$. I think that I cracked the real case (I got for $\ell^1$: $\{e_n\}_{n\in \mathbb ...
1
vote
1answer
52 views

If $f(x)=f(\delta x), \delta>0$ a.e then $f$ is constant?

Let $f\in L^{\infty}(0, \infty).$ For $\delta>0,$ $f(x)=f(\delta x)$ all most every where(a.e) on $(0, \infty).$ My Question: Can we expect $f$ is constant function on $(0, \infty)?$ If yes, how?
1
vote
0answers
35 views

Extreme point of unit balls, the complex case [duplicate]

I've been trying to determine what are the extreme point of the unit balls of $\ell^1$ and $C[0,1]$. I think that I cracked the real case (I got for $\ell^1$: $\{e_n\}_{n\in \mathbb ...
2
votes
1answer
33 views

If $f_n \to f$ in $L^p$, then prove that this sequence converges to $f$

Let $f_n \in L^p(\Omega)$ be a sequence that converges to $f$ in $L_p(\Omega)$. If $\Omega_n$ is a subset of $\Omega$ such that $\displaystyle{\lim_{n\rightarrow \infty}}\Omega_n=\Omega$, prove that ...
1
vote
1answer
28 views

How to prove that operators are isometry on $\ell^p$/$c$?

Lately I've been studying Banach spaces and isometries, and encountered many explicit isometrys involving $c$, $c_0$, $c^*$, $c_o^*$, $\ell^1$, $\ell^\infty$, etc... ($c \subset \ell^\infty$ is the ...
2
votes
0answers
24 views

Approximation of $L^2$ function by smooth functions on a manifold

Let $M$ be a $C^2$ compact Riemannian manifold with boundary. Suppose $f \in L^2(M)$ is such that $0 \leq f \leq 1$. Is it possible to find $f_n \in C^\infty(M)$ such that $0 \leq f_n \leq 1$ for ...
-3
votes
1answer
47 views

I don't know the answer [closed]

Let we have the sequence of real numbers $\left(\frac1{n^{1/n}} -1\right)$ clearly this sequence converges to zero , but I think it is not in any Lp space .
5
votes
1answer
63 views

Convergence of sequence of $L^{p}$ function

Given that $\Omega \subset \mathbb{R}^{n}$ is bounded. If you are given that $u_{k} \rightarrow u$ in $L^{p- \epsilon}(\Omega)$ and a functions $f: \mathbb{R} \rightarrow \mathbb{R}$ where ...
1
vote
1answer
19 views

Rapid decay times polynomial decay is L^1?

I have a smooth function $f(x)$ which is of rapid decay (actually I know it's a Schwartz function), and I have another function $g(x)$ which behaves like a polynomial as $x \to \pm \infty$; that is $$ ...
4
votes
1answer
35 views

A property of sobolev spaces

Let $W^{k,p}(\Omega):=\{y\in L^p(\Omega) : D^{\alpha}y\in L^p(\Omega)$ for all $|\alpha|\leq k\}$ I want to prove now that: (1) $u \in W^{1,2}(\mathbb R)$ is equivalent to (2) $u \in L^2(\mathbb ...
0
votes
0answers
12 views

If $\nabla \cdot (|\nabla u|^{p-2}\nabla u) \in L^2$ what space is $u$ in?

Define $\Delta_p u = \nabla \cdot (|\nabla u|^{p-2}\nabla u)$. I want to know, if $\Delta_p u \in L^2(\Omega)$, then what space is $u$ in? I am having trouble figuring it out. Take $p=2$. Then ...
5
votes
1answer
44 views

An example of a function not in $L^2$ but such that $\int_{E} f dm\leq \sqrt{m(E)}$ for every set $E$

I am thinking about this problem: Let $f\in L^1 [0,1]$ to be a nonnegative function satisfied: $$\int_{E} f dm\leq \sqrt{m(E)}$$ for every measurable set $E\subset [0,1]$, Prove that $f\in ...
2
votes
1answer
31 views

A basic question on $L^p$ norm

Consider a probability space and $f_m$ be sequence of measurable functions a.s. converging to $f$. What can be said about the limit $$ \lim_{m\to \infty} \|f_m\|_m$$ where $\|.\|_p$ stands for the ...
2
votes
0answers
30 views

Showing inequalities for $l^p$ sequences

If I show that an inequality (e.g. Holder or Minkowski) holds for the $L^p$ space, then can I automatically conclude that the inequality also holds for $\ell^p$ sequences, just by integrating wrt. the ...
2
votes
0answers
39 views

Particular $L^p$ space

I am confusing some definitions. Suppose we have a Cauchy sequence $(f_n) \subset L^2(\Omega,C^0([0,1],\mathbb{R}))$, where $\Omega$ is a measurable space with measure $\mu$ and ...
1
vote
0answers
29 views

Equivalence of weak $L^p$ norms

I'm kind of new to the subject of weak $L^p$ spaces. The definition of the (quasi-)norm in weak $L^p$ ($p\in(0; \infty)\,$) over $\sigma$-finite measure space $(X, \mu)$ I use is $||f||_{L^{p, ...
0
votes
1answer
27 views

Show that $L^p(\mathbb{R}^d)$ spaces are not comparable one another.

I have to show that, in $\mathbb{R}^d$ with Lebesgue measure, the $L^p$ spaces are not comparable one another. More precisely, I want to show that given $p$ and $q$ such that $1\leq p<q\leq\infty$, ...
2
votes
1answer
35 views

Convergence of truncation in $L^{p}$

If you have a truncation $T_{k}u$ defined as: $$ T_{k}u := \begin{cases} u,& \text{ if }~ |u(x)| \leq 1\\ k\frac{u}{|u(x)|}, & \text{ if }~|u(x)| > k \end{cases} $$ If you consider ...
1
vote
1answer
32 views

Does weak convergence in $L^{q}$ imply weak convergence in $L^{p}$

Assume we have $u_{k} \rightharpoonup u$ in $L^{q}(\Omega)$, does it then follow that $u_{k} \rightharpoonup u$ in $L^{p}(\Omega)$, given that $q > p$ and $\Omega \subset \mathbb{R}^{n}$ is ...
2
votes
1answer
50 views

estimate of infinite norm by $(p,q)$ norms

Let $p$ and $q$ be conjugate exponents, i.e. $\frac{1}{p}+\frac{1}{q}=1$. Prove or disprove: $$ \|f\|_\infty^2\le\|f\|_p\|f'\|_q $$ I think this is true. I tried to prove it using integration by ...
1
vote
1answer
31 views

Riesz Representation Theorem for $l_p$

Let $ 1 \leq p < \infty$, with $q$ the conjugate of $p$, and let $T \in l^{p*}$. Then for some sequence $g \in l^q,$ $T(f)=\sum_{\mathbb{N}} fg$ for all $f \in l^p$. I am trying to prove this ...
1
vote
2answers
40 views

If $F$ is the distribution function of an $L^p$ function, then $\lambda^p F(\lambda)\to0$ as $\lambda\to0$

Let $1\leq p<\infty$, $f\in L^p (\mathbb{R}^n)$. Let $F(\lambda)=m\{|f(x)|>\lambda\}$, show that: $$\lim_{\lambda\to 0} \lambda^{p}F(\lambda)=0$$ What I only know about distribution function ...
1
vote
1answer
45 views

Proving the Riemann-Lebesgue Lemma in $L^1(\mathbb{R}^n)$

$\mathbf{Riemann-Lebesgue \ Lemma \ in \ L^1(\mathbb{R}^n)}$. Suppose that $f \in L^1(\mathbb{R}^n)$. Then $\hat{f}(k) \rightarrow 0$ as $|k| \rightarrow \infty$. I cannot understand any of the ...
1
vote
1answer
25 views

Showing that $||\hat{f}||_{\infty} \leq ||f||_1$ in $L^1$

Let $f \in L^1(\mathbb{R}^n)$ then $\hat{f} \in L^{\infty}(\mathbb{R}^n)$ and $||\hat{f}||_{\infty} \leq ||f||_1$ How do you prove this or where can I find a proof of this fact?
1
vote
1answer
31 views

Closure of $l^1(\mathbb{N})$ in $l^2(\mathbb{N})$

I am trying to understand $l^p$ spaces better and I got stuck. I showed that $l^1(\mathbb{N})$ is a subspace of $l^2(\mathbb{N})$. I also found a counterexample which shows that $l^1(\mathbb{N})$ is ...
0
votes
1answer
26 views

Showing that $L^{\infty}([0,1])$ is not strictly convex

Can somebody give an example that shows that $L^{\infty}([0,1])$ (regarding $|| \cdot ||_{\infty}$) is not strictly convex? Thanks in advance!
1
vote
0answers
11 views

Estimate bounds on Minkowski distance from point to a segment in Lp space

Assumptions Let $L_p(x,y)=(\sum_i|x_i - y_i|^p)^{1/p}$ (Minkowski metric), $a,b$ be arbitrary $n$-dimensional points, $c$ be a point that satisfies $L_p(a,b) = L_p(a,c) + L_p(c,b)$, i.e., a point ...
0
votes
1answer
37 views

Example of a sequence in L1 with these conditions

Is there an example of a sequence $\{f_n\}$ in $L^1(\mathbb{R})$, such that: $\{||f_n||_1\}$ is bounded. There's a convergent subsequence $f_{\phi(n)}$, i.e. $\exists f \in L^1(\mathbb{R})$ such ...
1
vote
1answer
33 views

$\mathcal L^{\infty}$ space properties

Can anybody give an example that for $1 \leq p < \infty$ neither $\mathcal L^p (\mathbb R) \subseteq \mathcal L^{\infty} (\mathbb R)$ nor $\mathcal L^{\infty} (\mathbb R) \subseteq \mathcal L^p ...
1
vote
1answer
36 views

Does an integral operator with a symmetric integrable kernel have to be bounded on $L^2$?

Suppose $K(x,y)$ is a symmetric kernel. Let $\phi\in L^2(\Omega)$, where $\Omega$ everywhere is a domain in $R^n$. Can $\int_{\Omega}K(x,y)\,\phi(y)\,dy$ belong to $L^2$? In other words can an ...
0
votes
1answer
17 views

The norm of dual operator over $L^p(\Bbb R^N)\times L^p(\Bbb R^N)$

Let $1<p<\infty$ and $E:=L^p(\Bbb R^N)\times L^p(\Bbb R^N)$. Let $\Phi\in E^*$, i.e., the dual of $E$. Hence by Riesz representation we have there exist $u_0$, $u_1\in L^{p'}(\Bbb R^N)$ such ...
3
votes
1answer
60 views

Question on $L_p$ spaces involving $\lambda^n$-measure on $\mathbb{R}^n$

Q/ Consider $L_p=L_p(\lambda^n)$ with the Lebesgue measure on $\mathbb{R}^n$ and $1\leq p<\infty$. Let $f_0=|x|^{-\alpha}$ for $|x|<1$ and $0$ otherwise. Show $f_0\in L_p$ iff $p\alpha < n$. ...
1
vote
0answers
26 views

$L^2$ inequality for derivatives of polynomials on triangles

I'm reading a paper which states the following inequality, but the (presumably) elementary proof is cited to be in a document, which is too old to get access to. Let $p: \mathbb{R}^2 \to \mathbb{R}$ ...
3
votes
3answers
65 views

An example of a function in $L^1[0,1]$ which is not in $L^p[0,1]$ for any $p>1$

Title says most of it. Could you help me find an example? It is easy obviously to show a function that would not be in $L^p[0,1]$ for a specific $p$ (say $(1/x)^{1/p}$, but I can't see how it would ...
0
votes
1answer
36 views

Uniform lower bound on convex functions bounded in $L^2$ norm

Consider a class of (proper closed) convex function on $[0,1]^d$, which we shall denote $\mathcal{F}$. If every element of $\mathcal{F}$ is bounded in $L_2$, say $$\int_{[0,1]^d} |f(x)|^2\ dx\leq 1,$$ ...
2
votes
1answer
23 views

Are Schwartz functions in $L^{p}$ for $0 < p < 1$?

Let $S(\mathbb{R}^{d})$ denote the Schwartz functions in $\mathbb{R}^{d}$. I know that $S(\mathbb{R}^{d}) \subset L^{p}(\mathbb{R}^{d})$ for $1 \leq p < \infty$. Is $S(\mathbb{R}^{d}) \subset ...
0
votes
0answers
12 views

$L^p$ is a quasi normed space for $0<p<1$ [duplicate]

I know that $L^p$ is a vector space for $p>0$ and a normed space for $p \geqslant 1$ now I need show that for $ 0<p<1$ and $f,g \in L^p$ exist $K \in \mathbb{R}$ such that $||f+g||_p ...
3
votes
1answer
23 views

$L^\infty(S^1)$ is not separable

Let $S^1$ be the unit circle and $L^\infty(S^1)$ the space of measurable functions $f:S^1\to\mathbb{C}$ such that $\|f\|_\infty<\infty$. (In fact $L^\infty(S^1)$ consists of equivalence classes of ...