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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 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 problem is a $L^p$ space.

3
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1answer
43 views

Representation of linear operator between $L^p$ spaces.

I was wondering where I could find a reference to the fact that continuous linear operators: $$T:L^p(X,\mu)\to L^q(Y,\eta)$$ are of the form $T(f)(y)=\int_{X} k(x,y)f(x)d\mu$ for some $k$ satisfying ...
3
votes
1answer
35 views

Proving an inequality of random variables

I am currently reading a paper that claims the following fact: Let $X$, $Y$ be $L^2$ random variables on some probability space. The $L^2$ norm is denoted by $\| \cdot \|_{2}$. Then there exists $C&...
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vote
0answers
50 views

Show that $(f_n)_n$ is relatively compact in $L^p$ space

Let $I=[0,1]$, $Q=I\times I$ and $(u_n),(v_n)$ bounded sequences in $L^2(I)$. Assume $x\mapsto u_n(x), x\mapsto v(x)$ are continuous and monotone non decreasing on $I$ for all $n\in\mathbb{N}$; define\...
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1answer
39 views

Have I understood Compact Set correctly

In our current Measure Theory Class, we bought up the notion for a function $f:\mathbb R \to \mathbb R$ that is continuous to have a compact support, is equivalent to the fact that $\overline{\{x \in \...
1
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1answer
19 views

Convergence in $\ell^p$

Let $0\neq x=(x_1,x_2,\cdots,x_n,\cdots) \in\ell^4$. For which of the following values of $p$, the series $$\sum_{i=1}^\infty x_iy_i$$ converges for every $y=(y_1,y_2,\cdots,y_n,\cdots) \in\ell^p $ ? ...
1
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1answer
30 views

Bartle The Elements of Integration exercise 6S

Let $f_n$ $\in$ $L_p(X,\chi,\mu)$, $1 \leq p <+\infty$ and let $\beta_n$ be defined for $E \in \chi$ by $$\beta_n(E) = \left(\int_{E} |f_n|^p d\mu\right)^{1/p}$$ and suposse that $(f_n)$ is a ...
3
votes
1answer
22 views

Help bounding a “norm”

In Weak Convergence and Stochastic Processes, the authors introduce the following notation: $$\|\xi\|_{2,1} = \int_0^\infty \sqrt{P(\xi > x)}\,\mathrm dx$$ They then admit that this is technically ...
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1answer
33 views

Is the squared difference between a $L^{2}$-function and a Non-$L^{2}$-function in $L^{2}$?

Let $(\Omega, \mathcal{A}, P)$ be a probability space. Furthermore, let $Y,V : \Omega \rightarrow \mathbb{R}^n$ be random vectors and let $$ \int_{\Omega}{\vert \vert Y \vert \vert ^2}dP < \infty$...
1
vote
1answer
21 views

Show that if $p > 1$ and $\sup_{n \geq 1} |X_n| \in L^p$ then $\{X_n, n \geq 1\}$ is uniformly integrable.

Let $(\Omega,\mathcal{\Sigma,\mathbb{P}})$ be a complete probability space. I have to show that $$\lim_{\alpha \to \infty} \sup_{n \geq 1}\int_{\{|X_n| \geq \alpha\}}|X_n|\, d\mathbb{P} = 0$$ ...
2
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2answers
25 views

Bounded sequence in $\mathcal{L}^p$ implies $\mathcal{L}^p$ convergence if sequence converges a.e

I'm working on $\mathcal{L}^p(E,\mathcal{A},\mu)$ space and I would like to prove the following: Let $p\in [1,\infty[$ and $\mu(E)< \infty$, suppose also that i) $f_n \to f, \mu$ a.e ii) there ...
2
votes
2answers
30 views

Bounded sequence perpendicular to dense subset of $\ell^2$

Consider the real Banach space $\ell^2$ of square summable sequences and let $\mathcal{A}\subset \ell^2$ be a dense subspace. Suppose I have a bounded sequence $\psi=(\psi_n)_{n\geq 1}\in \ell^\infty$ ...
4
votes
1answer
62 views

Prove weighted ball in $l^2$ space is compact

Let $B \subseteq l^2$, $B=\left\{x\in l^2:\sum_{n\geq1}n|x_n|^2\leq1\right\}$, show that $B$ is compact. My thought: $B$ is closed in $l^2$ which is complete. Then $B$ is complete. It suffices to ...
1
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1answer
33 views

Compactness in weak$^*$-topology on $l_1(\mathbb{N})$

Let $K$ denote the closed unit ball of $l_1(\mathbb{N})$ (considered as a vector space over $\mathbb{C}$). Is $\mathrm{co}(\mathrm{Ext}(K))$ (the convex hull of its extreme points) compact in the ...
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0answers
33 views

Is the Jacobian different for different $l^p$ norms?

Because the Jacobian is related to the measure of an integral, and the measure is related to the norm/metric of the space, does the Jacobian behave differently for $l^p$ spaces where $p$ isn't $2$ ...
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0answers
30 views

Finding a norm 1+epsilon projection onto a subspace of Lp

Hey guys I have a quick question. Let $\mu$ be a probability measure and let $E$ be a closed subspace of $L_p(\mu)$, $1\leq p\leq \infty$ which has either finite dimension or codimension. For any $\...
2
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0answers
36 views

Inclusion relationship between $\mathscr l^p$ spaces

In Folland’s Real Analysis book, we have the following proposition: “ If $\mu(X)< \infty$ and $0<p<q\le \infty$, then $L^p(\mu)\supset L^q(\mu)$.” So suppose that $A$ is a finite set and $\...
1
vote
1answer
42 views

Prove boundedness of operator on $L^p$

Let $\alpha \in (0,1)$, $p\in(1,\infty)$ and $T >0$ The operator is defined as $I^\alpha f(t) = \frac{1}{\Gamma(\alpha)}\int_{0}^{t} (t-r)^{\alpha-1}f(r)dr$. I want to prove that $I^\alpha \in B(...
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0answers
19 views

Smooth endpoint map

Consider a control system $$ \dot x(t) =f(x(t),u(t)) \qquad (\star) $$ where $f$ is a smooth vector field and $x\in \mathbb{R}^n$. The endpoint mapping is defined by $$ E:\mathbb{R}^n\times \mathbb{...
1
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1answer
47 views

Prove that the weak convergence in $L^p \left(\mathbb{R}^N\right)$ does not imply the weak convergence of the modulus

Let be $p\in [1,\infty)$ and $\{u_n\}\subset L^p\left(\mathbb{R}^N\right)$ a sequence. How do I prove that $u_n \rightharpoonup u$ in $L^p \left(\mathbb{R}^N\right) \not\Longrightarrow|u_n|\...
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2answers
22 views

Example for $f \in \mathcal{L}^q(\mathbb{R}, \lambda) \setminus \mathcal{L}^p(\mathbb{R}, \lambda)$ for $q < p$

We consider $q, p \in [1, \infty)$. I'm required to show that $\mathcal{L}^p(\mathbb{R}, \lambda) \subsetneq \mathcal{L}^q(\mathbb{R}, \lambda)$ and $\mathcal{L}^q(\mathbb{R}, \lambda) \subsetneq \...
5
votes
1answer
68 views

Is the $p$-norm ever a norm for $0<p<1$?

I wonder: Is there a measure space $(X,\Sigma,\mu)$ such that $L^p(\mu)$ form a normed space w.r.t the $p$-norm, for some $0<p<1$?(assuming that $X$ contains more than point). I know that in ...
1
vote
2answers
43 views

Vitali's Convergence Theorem but one hypothesis changes

We have the following problem: Let $(Y, \Gamma , \nu)$ be a measure space. Suppose that $\{g_{n}\} \, \subset \, L^{p} \, := \, L^{p}(Y,\Gamma , \nu).$ Prove that $\lim_{n} g_{n} = g$ in $L^p$ if ...
2
votes
1answer
44 views

Minimizing $\ell_p$ distance to affine subspace of affine subspace

Suppose that $V$ is a finite-dimensional real vector space equipped with some $\ell_p$ norm for $1 < p < \infty$. ($p$ is chosen this way so as to be strictly convex.) Then suppose we have an ...
3
votes
2answers
67 views

A proposition about bounded and weak-$L^2$ integrable fucntion

I have this proposition that I have no idea how to start. any hints helps. If $f\in L^{2,\infty}\cap L^{\infty}$ then $f\in L^p$ for $p\geq2$ and the second part asks me to generalize this result. ...
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0answers
54 views

Riesz lemma for $L^p$ space

I need a proof for special case of Riesz lemma (when $\varepsilon$ is 0): If Y is a closed proper subspace of $L^p(\mu)$ for some $1<p<\infty$, then there exist $f\in L^p(\mu)$ such that $||f||...
1
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1answer
25 views

Alternative form for Liapunov inequality

Let $1<p<q<\infty$, and $r\in [p,q]$ whith $\frac{1}{r}= \frac{\alpha}{p}+ \frac{1-\alpha}{q}$. If $f\in L_p\cap L_q$ then $$\|f\|_r \leq \|f\|_p^\alpha\|f\|_q^{(1-\alpha)}$$ My teacher ...
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0answers
45 views

Brezis 4.15, $f_n(x)=ne^{−nx}$

Let $\Omega=(0,1)$ and $f_n(x)=ne^{-nx}$ prove that (i) $f_n\rightarrow0$ a.e. (ii) $f_n$ is bounded in $L_1(\Omega)$ I know that for $f_n$ to converge a.e. to zero, the set of points in which this ...
1
vote
2answers
57 views

Expectation of $XY$ bounded for all bounded $Y$ implies $X$ is $L^p$

I'm trying to prove: Let $X$ be a real random variable, $p, q \in (1,\infty)$, $\frac 1 p + \frac 1 q = 1$. If there is $C < \infty$ such that $|\mathbb E[XY]| \leq C ||Y||_q$ for any bounded ...
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0answers
25 views

Brezis excercise 4.12: $L_p$ is uniformly convex for $1<p\leq 2$

Let $1<p<\infty$. Prove that there is a constant $C$(depending only on $p$) such that $$|a-b|^p\leq C(|a|^p+|b|^p)^{1-s}(|a|^p+|b|^p-2|\dfrac{a+b}{2}|^p)^s$$ for all $a,b\in \mathbb{R}$ and $s=\...
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1answer
53 views

Excercise 4.3 (3) in Brezis. Convergence in $L_p$

Let $(f_n)$ in $L_p(\Omega)$ $1\leq p< \infty$ and $(g_n)$ bounded in $L_{\infty}(\Omega)$ assume that $f_n \rightarrow f$ in $L_p(\Omega)$ and $g_n \rightarrow g$ a.e. Prove that $$f_ng_n\...
1
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1answer
23 views

For which $p \in [1,\infty]$ does $g, g_{A}$ and $g_{A^{c}} \in \mathcal{L}^p$ hold?

Let $s \in \mathbb R, d \in \mathbb N$, $g:\mathbb R^{d}\to \bar{\mathbb R}, x \mapsto |x|^s$ . Set $E_{d}:=\{x\in \mathbb R^{d}:|x|\leq 1\}$ Determine $p \in [1,\infty]$ for which $g, g\chi_{E_{d}}$...
1
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1answer
21 views

analysis of $T : f \to Tf$ with $[T(f)](x) = ie^{i\pi x}(\int_0^x e^{-i\pi t}f(t)dt - \int_x^1 e^{-i\pi t}f(t)dt)$

$\, f \in L^2([0,1],\mathbb{C})$ show that $T : f \to Tf, \, f \in L^2([0,1],\mathbb{C})$ is continuous, $[T(f)](x) = ie^{i\pi x}(\int_0^x e^{-i\pi t}f(t)dt - \int_x^1 e^{-i\pi t}f(t)dt)$ the ...
2
votes
0answers
10 views

$C_0(\Delta(l^p))$ description [duplicate]

If $X=l^p$ ,$p \in [1,\infty)$ is nonunitial Banach algebra with coordinate multuplication and $\Delta(l^p)=\{e_n:n \in \mathbb{N}\}$, where $e_n(x)=x_n$ for $x \in l^p$, what is $C_0(\Delta(l^p))$ ?
1
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1answer
16 views

Brezis excercise. Does $h\in L_p(\Omega)$?

Given two functions $f,g\in L_p(\Omega)$ define $h(x)=max\{f(x),g(x)\}$ prove that $h\in L_p(\Omega)$. I know that $|h(x)|^p\geq |f(x)|^p$ and $|h(x)|^p\geq |g(x)|^p$ but this seems to go nowhere... ...
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0answers
34 views

Brezis exercise, under what condition does $f$ belongs to $L_p(\mathbb{R^n})$

Brezis excercise 4.1: I am asked to show under what conditions does $f(x)=\{1+|x|^\alpha\}^{-1}\{1+|\log|x||^\beta\}^{-1}$ belong to $L_p(\mathbb{R}^n)$ where $\alpha,\beta>0$.. I know that for $f$...
3
votes
2answers
61 views

What is the operator norm of $Tf(x) = x^2f(x)$?

Let $H = L^2([0,1],\mathbb{R})$ and $T : H \to H,\, Tf(x) = x^2f(x) $. $T$ is linear. $$\|Tf\|_{L^2([0,1],\mathbb{R})} = \sqrt{\int_0^1x^4f^2(x)dx} \leq\sqrt{\int_0^1f^2(x)dx} = \|f\|_{L^2([0,1],\...
2
votes
0answers
31 views

Banach algebra $l^p$ is not isomorphic to $C^{*}$ algebra

Consider commutative Banach algebra $l^p$, $p \in [1,\infty)$ with multiplication by coordinates. I know, that $\Delta (l^{p})=\{e_n : n \in \mathbb{N}\}$ - set of canonical functionals. We know that $...
1
vote
2answers
48 views

Dual space of $L^p(\Omega,\mathcal{A},\mu,\mathbb{R}^d)$.

I want to show that for $p\in(1,+\infty)$ the dual space of $L^p(\Omega,\mathcal{A},\mu,\mathbb{R}^d)$ is isometrically isomorphic to $L^q(\Omega,\mathcal{A},\mu,\mathbb{R}^d)$, where $\frac{1}{p}+\...
3
votes
1answer
43 views

Interchanging limit and integral.

Suppose $(X,\mu)$ is a probability space, $W\in L^1(X)$, $V\in L^\infty(X)$, and $V_n\to V$ in $L^2(X)$ (in my situation $V_n$ is the partial Fourier sum and so the $L^2(X)$ convergence is automatic). ...
0
votes
0answers
34 views

Is it the case that $\|f\|_{L^p (\mathbb{R})} < \infty$ iff $\|f\|_{L^p (\mathbb{R})}^p < \infty$ for all $1 \leq p < \infty$?

Is it the case that, for all $1 \leq p < \infty$, $$\|f\|_{L^p (\mathbb{R})} < \infty \quad \text{if and only if} \quad \|f\|_{L^p (\mathbb{R})}^p < \infty?$$ When computing $L^p$ norms, I ...
1
vote
1answer
17 views

$f \in C_{00}(\mathbb{R^p},\mathbb{C})$. $ \mapsto f_t \in L_\infty(\mathbb{R}^p, \mathcal B_p, \lambda_p, \mathbb{C})$ uniformly continuous?

Continuing from here Let $f_t(x):=f(x+t)$ Consider $f \mapsto f_t$ which is a linear, isometric bijection from $L_\infty(\mathbb{R}^p, \mathcal B_p, \lambda_p, \mathbb{C})\to L_\infty(\mathbb{R}^p, \...
-1
votes
1answer
38 views

$y \in \mathbb{K^N}$ a scalar sequence, if $\sum_{n \geq 1} x(n)y(n)$ is bounded,does $x \in l_{p^*}$? [closed]

Given $y \in \mathbb{K^N}$ a scalar sequence, and $1<p<\infty$. Suppose that, $\forall x \in l_p$, the series $\sum_{n \geq 1} x(n)y(n)$ is bounded. Show that $x \in l_{p^*}$. With $l_{p^*}$ ...
3
votes
1answer
48 views

Given that $f \in L^2(\mathbb{T})$ and the sequence of Fourier coefficients $(\hat{f_n})\in l^1(\mathbb{Z})$, must $f$ be continuous?

Note that $\mathbb{T} = \{ z \in \mathbb{C} : |z| = 1 \}$. This detail wouldn't fit in the title. This is a previous exam question I am practicing with and I'm at a loss! Any advice on how to think ...
0
votes
1answer
39 views

Show that $\phi(r) = ||f||_{L^r(0,1)}$, $r\in [1,2]$ is continuous function of r.

Show that the function $$\phi(r) = ||f||_{L^r(0,1)}$$ for $r\in [1,2]$ is a continuous function of $r$. Do I need to use the definition of continuity to solve this question? I don't know how to start ...
1
vote
1answer
38 views

Simple bound for $L^p$ norm

Is there a bound for any $1<p<\infty$ or specifically $p=6$ such that $$||u||_{L^{p}(U)}\leq C ||u||_{H^{1}(U)} $$ Where $U$ is an open bounded set of class $C^2$ in $\mathbb{R^3}$ and $H^{...
-1
votes
1answer
31 views

Find $f\in L^p(0,1)$, but $f\notin L^q(0,1)$, $p < q$

I want to find a measurable function $f \in L^p(0,1)$, for $p\in [1,+\infty)$, but $f \notin L^q(0,1)$ for each $q\in (p,+\infty]$. I tried to manipulate $f=\frac{1}{x^a}$, improving the exponent $a$ ...
2
votes
1answer
34 views

For which values of $c$ is the function $f(x)$ in $W^{1,p}_{\mathrm{loc}} (\mathbb{R})$?

We're given the function $$f(x) = \begin{cases} 2 \sin(x) + 3, & x> 0 \\ -2 \sin(x) + c, & x \leq 0 \end{cases} \\$$ and asked to find the values of $c$ for which $f \in W^...
0
votes
1answer
35 views

Function in $L^2$ that doesn't vanishing

Is this statement makes a sense: $f\in L^2(0,1)$ such that $f(x)\ne 0 ,\forall x\in (0,1)$ ?
1
vote
1answer
30 views

Showing $\mathbb{R}^p\ni t \mapsto f_t \in L_\infty(\mathbb{R}^p, \mathcal B_p, \lambda_p, \mathbb{C})$ is not continuous

Let $f_t(x):=f(x+t)$ I want to show that $f \mapsto f_t$ is a linear, isometric bijection from $L_\infty(\mathbb{R}^p, \mathcal B_p, \lambda_p, \mathbb{C})\to L_\infty(\mathbb{R}^p, \mathcal B_p, \...
2
votes
1answer
71 views

Is the set of sequences $(x_n)$ in $\ell^1$ such that $|x_n|\leqslant\frac1n$ for every $n$, compact in $\ell^1$?

I was thinking about proving a set $S$ is compact in $l^{1}$ space. $S = \{f \in l^{1} : |f(k)| \leq \frac{1}{k} \forall k \}$ where $\{f_{n}\}_{n=1}^{\infty}$ be functions such that $f_{n} : \Bbb{N}...