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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24 views

Relation between $L^{p}$ norm of derivative $\|Dg\|_{L^{p}}$ and $\|g\|_{L^{1}}$

Let $\phi \in \mathcal{S}(\mathbb R)$(Schwartz space) with $\phi =1$ on $[-1, 1].$ Put $g:=\phi^{\vee}$(inverse Fourier transform of $\phi$). My Question: Can we show ...
2
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1answer
78 views

Characterizing when composition by a power function lies in an $L^p$-space

This is a past qual question that I have been struggling with: let $p,q,r \geq 1$. One would like to characterize the constants $q$ such that $f(x^r) \in L^q ((0,1))$ for all $f \in L^p((0,1))$, that ...
2
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1answer
56 views

Which is finer(larger) between the sequence spaces $l_{p}$ & $l_{p+1}$

Prove that, $l_{3}\subset l_{7}$ & $L_{9}[0,1]\subset L_{6}[0,1]$, where $l_{p}$ & $L_{p}[0,1]$ are of usual notation. Are the converses hold for both cases? Can these two results ...
3
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1answer
84 views

A function $f$ such that $f \in L_1$ but $f \notin L_p$ for $p>1$ [duplicate]

I want find a function $f: [0,1] \mapsto \mathbb{R}$ such that $f \in L_1[0,1]$ but $f \notin L_p[0,1]$ for all $p>1$. My attempts: First I thought in the family of functions $\frac{1}{x^\alpha}$ ...
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1answer
36 views

Is $\left(\frac{1}{x}\right)^{\frac{1}{p} - \frac{1}{n}}$ a Cauchy Sequence in $L^p((0,1))$

Is $(\left(\frac{1}{x}\right)^{\frac{1}{p} - \frac{1}{n}})_{n\in N}$ a Cauchy Sequence in $L^p((0,1))$? and does it converge to $\frac{1}{x}^{\frac{1}{p}}$ (p is a real number bigger or equal to 1) I ...
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0answers
56 views

Dual of $L^p$ when $p = 0$?

I've spent some time searching for this online - both on this site and elsewhere - and even after consulting a considerable amount of literature, I can't seem to nail down an answer. Perhaps someone ...
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2answers
28 views

A basic question on the space of square integrable functions

I have seen in a book the following cliam: Let $f_m,f \in L^2[0,N]$ and $\frac{1}{m}\sum_{k=1}^{m}f_{n(k)} \to f$ in $L^2[0,N]$ for a subsequence $n(k)$ Then for any $g \in L^2[0,N] s.t. \|g\|=1$ ...
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1answer
30 views

Find $c_0$ for which a sequence is in $l_2$?

Let $y \in l_2$ and let $$x_k=\left[C_0+\sum\limits_{j=0}^{k-1}\frac{y_j}{(\lambda+1)^{j+1}}\right](\lambda+1)^k.$$ Does there exits unique constant $C_0$, such that $x \in l_2?$ I need to show the ...
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1answer
48 views

Generalised Holder ineq

Prove the following generalisation of Holder's inequality $$\int | u_1 \cdot ... \cdot u_N | d\mu \leq \|u_1\|_{p_1} \cdot ... \cdot \|u_N\|_{p_N}$$ for all $p_j \in (1,\infty)$ such that ...
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0answers
74 views

Dual of $l^p$ Direct sum

I am asked to show that the $l^p$-direct sum of a sequence of Banach Spaces $X_n$ is isometrically isomorphic to the $l^q$ direct sum of $X_n^*$ where $X_n^*$ is the dual of $X_n$ for each $n$ ...
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1answer
44 views

Integral with a compact supported function $0$ indicates the $L^2$ function $0$ almost everywhere

Suppose we have $f\in L^2([0,1])$,and for every $\varphi\in C_{0}^{\infty}((0,1))$, we have $$\int_0^1 f(x)\varphi(x)dx=0$$ Then how can I show $f=0$ a.e? I know when $f\in C^0([0,1])$ the results ...
5
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0answers
62 views

Alternate proof of a result on dual spaces: what is wrong with it?

I am familiar with Rudin's book's proof of the fact that, in $\sigma$-finite measure spaces and for $p\in[1,+\infty)$, the dual space of $L^p$ is $L^q$ where $p,q$ are conjugate, i.e. ...
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1answer
40 views

Show that $g=\sum_{n=1}^{\infty } |f _{n+1 }-f _n | $ has $||g ||_p\le 1 $ if $||f _{n+1 }-f _n ||_p <2 ^{-n } $

Minkowskis inequality implies that $g _k=\sum_{n=1}^{k} |f _{n+1 }-f _n | $ has norm less than $1 $, and there is a hint to use Fatou's lemma to $g _k ^p$. Then $\int \lim \inf g _k ^p \le \lim \inf ...
2
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1answer
33 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 ...
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0answers
28 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} ^ ...
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1answer
65 views

$L^p$ Martingale convergence theorem

I am trying to prove the $L^p$ Martingale convergence theorem for martingale $X=(X_n)^{\infty}_{n=0}$ on $(\Omega,\mathcal{F},(\mathcal{F}_n)^\infty_{n=0},\mathbb{P})$ which is bounded in $L^p$ for ...
2
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0answers
63 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
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1answer
79 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), ...
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0answers
31 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 ...
2
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1answer
62 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 ...
4
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1answer
88 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 ...
4
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1answer
57 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
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1answer
45 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
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1answer
38 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
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1answer
49 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 ...
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1answer
54 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
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1answer
61 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?
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0answers
39 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 ...
1
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1answer
49 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 ...
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1answer
41 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. (Here $c\subset\ell^\infty$ is the ...
2
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0answers
31 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 ...
5
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1answer
77 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 ...
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1answer
22 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
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1answer
51 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 ...
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0answers
16 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
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1answer
55 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
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1answer
43 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 ...
3
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1answer
39 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
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0answers
44 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 ...
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0answers
50 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
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1answer
34 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
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1answer
57 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 ...
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1answer
58 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
54 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
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1answer
47 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
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2answers
42 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 ...
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1answer
165 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 ...
2
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1answer
29 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
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1answer
34 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
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1answer
29 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!