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.

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12 views

Relation between the types of convergences of sequences in $\mathbb{L}_p$ Spaces.

As far as I know there are 4 types of convergence in $\mathbb{L}_p$ spaces. 1. Pointwise Convergent a.e 2. Uniformly Convergent almost a.e 3. Convergence in Measure 4. p-Convergence My question is how ...
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1answer
15 views

Existence of constant for a “Minkowski-like” inequality to hold on $L_p$ $p<1$.

I'm solving some problems to prepare for my phd qualifying exam on functional analysis and measure theory. I want to prove that given a measure space $(X,\mathcal{A},\mu)$ for every $0<p<\infty$...
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2answers
43 views

Convergence of double Integral

Let's assume that $f \in L^p(\mathbb{R})$ ($1 \leq p < \infty$). Does it then hold that $$ \lim_{n \rightarrow \infty} \int_0^1 \int_0^1 \left \lvert f\left( \frac{\tilde{r}}{n} \right) -f\left(\...
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2answers
27 views

Does $L^1$ imply $L^p$ on finite measure spaces?

If $(\Omega,\mu)$ is a finite measure space, i.e., if $\mu(\Omega)<\infty$, then does $f\in L^1(\Omega)$ imply that $f\in L^p(\Omega)$ for every $p$? This is just a statement that I feel like I've ...
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1answer
31 views

Riesz representation theorem vs. natural duality for $L^2$

We know that the spaces $L^p(\Omega)$ and $L^q(\Omega)$ are isometric and isomorphic for $p,q$ conjugate and $p,q \neq 1,\infty$. Call the isomorphism $l\colon L^p(\Omega) \to L^q(\Omega)$. Take $p=q=...
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1answer
24 views

extension of integral preserving positive operators on $L^p$ into $L^q$

Let $(\Omega,\mu)$ be a finite measure space. Let $T \colon L^\infty(\Omega) \to L^\infty(\Omega)$ be a weak* continuous contractive positive operator such that $\int_\Omega T(f)=\int_\Omega f$ for ...
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1answer
33 views

Identifying a generic Hilbert space $H$ with an $L^2$ space on some measure space.

This may be a stupid question, but I was wondering, if we are given an infinite dimensional Hilbert space $H$, is it possible to find (or to hypothesise that there's) a measure space $(M,\mathcal M,\...
6
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1answer
47 views

$f\in L^2[0,1]$ iff $f\in L^1[0,1]$ and there is nondecreasing $g$ with $|\int_a^b f(x)dx|^2 \leq (g(b)-g(a))(b-a)$ for $0\leq a\leq b\leq 1$

Let $f:[0,1]\to \Bbb C$ be measurable. I am trying to show that $f\in L^2$ iff $f\in L^1$ and there is a nondecreasing function $g:[0,1]\to \Bbb R$ such that $$ \left\lvert \int_a^b f(x)~dx \right\...
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1answer
15 views

multiplication of measurable functions in $L^p$ spaces

Let $(X, M, \mu)$ be a measure space, $q \in (0, +\infty]$ and $f,g : X \rightarrow \mathbb{C}$ in which $f \in L^{\infty} (\mu)$ and $g \in L^q (\mu)$. I want to show that $fg \in L^q (\mu)$. For ...
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1answer
22 views

Show $l^p$ is not complete with the $q$ norm

I know the question has been asked here, but I do not understand the solution (Are $\ell_p$ spaces complete under the $q$-norm?) I came up with my own solution and was wondering if it is correct. ...
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0answers
29 views

Understanding proof of Jensen's inequality in Lieb-Loss

I'm reading a book (Lieb–Loss) and in Section 2.2, they present a proof of Jensen's inequality and there's a step I don't quite understand. To set this up, suppose $J:\mathbb R\to\mathbb R$ is convex. ...
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22 views

Prove that $\Vert f \Vert_{L^p(\mathbb R^N)} > 0$ if $\Vert f \Vert_{L^2(\mathbb R^N)}, \Vert \nabla f \Vert_{L^2(\mathbb R^N)} > 0$

How can I prove that $$\Vert f \Vert_{L^p(\mathbb R^N)} > 0$$ for all $p \ge 2$ if $$\Vert f \Vert_{L^2(\mathbb R^N)}= \alpha > 0 \qquad \text{ and } \qquad \Vert \nabla f \Vert_{L^2(\mathbb R^N)...
2
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1answer
42 views

$L^\infty(\mathbb{R}^n)$ function that is also homogenous with degree zero

Consider a homogeneous function $m$ in $\mathbb{R}^n$ with degree zero, ie $$m(\lambda \xi) = m(\xi), \;\;\;\;\;\; \forall \lambda >0.$$ Is it true that $m \in L^\infty(\mathbb{R}^n)$ if, and only ...
2
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1answer
23 views

can sequences in $\ell^p$ with $0<p<1$ be multiplied by $n^\alpha$ and still fall in $\ell^1$?

(The following question is equivalent to the title) Let $(x_n)\in \ell^1$ be a real sequence, and let $1<p<\infty$. There exists an $\alpha>0$ such that the sequence $(n^{\alpha}x_n^p)$ is ...
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1answer
26 views

Fundamental set in the space of bounded sequences

Definition: Set $S$ fundamental set in Banach space $X$ if $\overline{Lin(S)}=X$. If $e_n=(0,\ldots ,0,1,0,\ldots)$ is a sequence that has $0$ everywhere, except on the $n$-th place and $e=(1,1,1,\...
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61 views

Can $\frac{\|f(x)\|^2}{2\|xf(x)\|}$ be written in terms of some mix of differential operators?

Can $$\frac{\|f(x)\|^2}{2\|xf(x)\|}$$ be written in terms of any mix of differential operators, be it the gradient, or Jacobian, or Laplacian, or somesuch? The term can be written as $$\frac{\left(\...
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0answers
35 views

Prove the relative compactness of a sequence in $L^p$

Let $I=[0,1] \subseteq \mathbb{R}$ and let $(u_n),(v_n)$ be sequences in $C(I)$ such that $$|u_n(0)|+|v_n(0)| \le 1,~~~~ |u'_n(t)|+|v'_n(t)| \le t+e^t ~~~~ \forall t \in I, ~\forall n.$$ I would like ...
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1answer
19 views

If $X_n$ converges to $X$ in $L_p$ and $Y_n$ converges to $Y$ in $L_p$ then $X_n + Y_n $ converges to $X + Y$ in $L_p$ [closed]

I want to show that if $X_n \xrightarrow{L^p} X$ and $Y_n \xrightarrow{L^p} Y$ then $X_n + Y_n \xrightarrow{L^p} X + Y$ ($p \geq 1)$. My idea is to use the following facts (whose proofs I won't give ...
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15 views

extrapolation and Stein's interpolation

Here $S=\{z \in \mathbb{C} : 0 < \Re z < 1\}$, $\overline{S}=\{z \in \mathbb{C} : 0 \leq \Re z \leq 1\}$ are the open and the closed strip and $(\Omega, \mu)$ is a probability space. Let $(T_z)_{...
3
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0answers
37 views

$L^2$ norm of inverse differential operator

This has come up in Lemma 1 of Mandache's 2001 paper on exponential instability for the inverse problem of the Schrodinger operator. Let $\Omega = B(0,1)$ in $\mathbb{R}^d$. Suppose $r_0\in (0,1)$ and ...
3
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1answer
57 views
+50

Twice continuously differentiable function $f:\Bbb R^2\to \Bbb R$ such that $f, f_{xx}, f_{yy}\in L^2$ is bounded

Suppose $f:\Bbb R^2\to \Bbb R$ is twice continuously differentiable, and that $f, f_{xx}, f_{yy}\in L^2$. Then is it true that $f$ is bounded? I first tried to find a counterexample, but I couldn't. ...
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2answers
22 views

Show that convergence in probabiltiy plus domination implies $L_p$ convergence

I want to show that if random variable $X_n $ converges to $X$ in probability (Let $(\Omega, \mathcal{A},P)$ be the probability triple) and $|X_n| < Y \,\,\forall\, n$ then $X_n$ converges to $X$ ...
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1answer
18 views

inverse of the modulus in $L^p$

We all know that, given any bounded open set $\Omega\subset \mathbb R$ $$\frac{1}{x}\in L^p(\Omega)\iff p<1 \qquad \frac{1}{x}\in L^p(\Omega^c)\iff p>1$$ what do these conditions become for a ...
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1answer
64 views

Question about $\lim _{q \rightarrow \infty}\|f\|_{q}=\|f\|_{\infty}$

Let $(X,B,\mu)$ be a complete measure space,Show that $$\lim _{q \rightarrow \infty}\|f\|_{q}=\|f\|_{\infty}, \quad \forall f \in \bigcup_{p} \bigcap_{p \leqslant q<\infty} L^{q}$$ So,$\lim _{q \...
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2answers
59 views

Show that there exist $f_1 \in L^{p_1}$ and $f_2 \in L^{p_2}$ such that $f = f_1 +f_2$.

Let $1 \le p_1 < p \le p_2 \le \infty$ and let $f\in L^p$. Show that there exist $f_1 \in L^{p_1}$ and $f_2 \in L^{p_2}$ such that $f = f_1 +f_2$. Just for clarify, we consider the $L^p$-spaces on $...
1
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1answer
65 views

Introduction to $L^2$ space. Equivalence class concept

Given a probability space $\left(\Omega\text{, }\mathcal{F}\text{, }\mathbb{P}\right)$, let $L^2$ denote all (equivalence classes for a.s. equality of) random variables $X$ such that $\mathbb{E}\{X^2\}...
4
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1answer
35 views

Is this norm equivalent to the $\ell_1$ norm?

I am studying for my qualifying exams and was asked to prove or disprove that the following norm is equivalent to the $\ell_1$ norm: $$\lVert x \rVert' = 2\left\lvert \sum_{n=1}^{\infty}x_n \right\...
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1answer
26 views

Convergence of the sum constructed by approximation of the integral of an $L^1$ function

Let $f \in L^1( \mathbb{R}^n)$. Does the sum $$S(x) = \sum_{k \in \mathbb{Z}^n} f(k+x) $$ converge for almost every $x$? Intuitively I'm approximating the integral (which is finite), so I think this ...
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2answers
31 views

Weak convergence definition

I have a following question, for example: let $f_n\in L^2$ be a bounded sequence of real functions. Then we know that there is some $f$ (in $L^2$?) such that $f_n$ converges weakly to $f$ in $L^2$ i.e....
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1answer
31 views

$L^{\infty}(\mathbb{R}^{N})$ and smallness of function

Let $(f_{n})_{n\in\mathbb{N}}\subset C(\mathbb{R}^{N})$ be a sequence of real valued function such that $\|f_{n}\|_{L^{\infty}(\mathbb{R}^{N})}\to\infty$ as $n\to\infty$. Then, I know that there ...
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3answers
53 views

about properties of the operator $T_f(g) := f\cdot g$.

let $f \in C([0,1])$ and $T_f: L^2([0,1]) \to L^2([0,1])$ and $T_f(g) := f\cdot g$ prove : 1)$T_f$ is well define , linear and bounded and find $|| T_f ||$ . 2 )if $T_f$ be compact operator then $f=...
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0answers
17 views

Prove that if $|f(t)-g(t)|^{2}_{L^{2}(\Omega)}\to 0,$ then $f(t)-g(t)\to 0\; a.e.$ in $\Omega \times (t_{0}, +\infty)$ as $t\to \infty$

Let $f, g$ be real-valued functions defined on $\Omega\times (t_{0}, +\infty),$ where $\Omega$ is a bounded domain in $\mathbb{R}^{d}\;(d\geq 1) with \;smooth \;boundary, t_{0}\in \mathbb{R}.$ ...
2
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1answer
29 views

Need an Upper Bound for $L^2$-Norm of Integral of a Gauss Function in 2 Dimensions

Statement of the Problem We wish to show that the following norm: $ \large || \int^{t/2}_{0} \xi_1 e^{-(t-s)|\xi|^{\alpha}} \int_{\eta \in \mathbb{R}^2} \frac{|\xi|^2 \eta_2 e^{-(S+1)|\xi - \eta|^{\...
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1answer
31 views

Rudin's Real and Complex Analysis, Section 9.16

In Section of 9.16 from Rudin's RCA, it says Let $\hat{M}$ be the image of a closed translation-invariant subspace $M \subset L^2$, nder the Fourier transfrom. Let $P$ be the orthogonal projection of ...
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1answer
39 views

Changing norm between $\ell^2$ and $\ell^\infty$.

For two finite sets $A$ and $B$, let $x_{i,j}$ be in both $\ell^2(A)$ and $\ell^\infty(B)$. Then is it possible that $$ \| \|x_{i,j} \|_{\ell^\infty(B)}\|_{\ell^2(A)} = \| \|x_{i,j} \|_{\ell^2(A)}\|_{\...
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0answers
28 views

Using a basis for $L^2(\mathbb{R}^d)$ to get a basis for $L^2(\{x\}\times\mathbb{R}^{d-k})$ for all $x$

I am in the situation where it would be very convenient if I could take a basis $\{f_i\}_{i = 1}^\infty$ for $L^2(\mathbb{R}^n)$ and manipulate it in some way to get a basis for the square-integrable ...
0
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1answer
44 views

Why $L^2\cap L^p $ is not dense in $L^{\infty}$?

Ein Euclidean space $\mathbb{R}^n$. Why $L^2\cap L^p$ is not dense in $L^{\infty}$? I have that $L^2\cap L^p$ is dense in $L^p$ with $1\leq p<\infty.$ Indeed, for $g\in L^p$ with $1\leq p<\infty$...
3
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1answer
47 views

Is $C_0$ dense in $l^{\infty}$

Is $C_0$ dense in $l^{p}$ with $1\leq p\leq \infty$ where $C_0=\{ (x_n): x_n\rightarrow 0, x_n\in R\}$. Well I think that if $p<\infty$ is true because by definition if i take $y=(y_n)\in l^p$ then ...
1
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2answers
30 views

Topology of $p$-integrable functions space.

In a reference, I read that topology of $L^p(\mathbb{R}^n)$, $1\leq p\leq \infty$. What is the topology of $L^p(\mathbb{R}^n)$? I know that $f\in L^p$, $\|f\|_{p}^{p}=\int_{\mathbb{R}^n}|f(x)|^{p}dx$. ...
1
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2answers
54 views

$\lim_{n\to \infty}(\int_0^1 f(x)^{2n}g(x)^n h(x)~dx)^{1/n}$ where $f,g,h$ are positive continuous functions on $[0,1]$

I want to find $\lim_{n\to \infty}(\int_0^1 f(x)^{2n}g(x)^n h(x)~dx)^{1/n}$ where $f,g,h$ are positive continuous functions on $[0,1]$. By Holder's inequality, this limit is greater than or equal to $...
0
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0answers
13 views

Belonging VS converging in a LP space

I'm reflecting on the meaning and the differences between belonging and converging to a $L^p$ space for a sequence of functions. Let's take this sequence as an example (please correct anything that is ...
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0answers
13 views

If $u \in L^2(0,T;L^2(\Omega))$ with $u(t) \in L^\infty(\Omega)$ uniformly, is $u \in L^\infty(0,T;L^\infty(\Omega))$?

Suppose I have a function $u \in L^2(0,T;L^2(\Omega))$. If I know that for almost all $t$, $\lVert{u(t)}\rVert_{L^\infty(\Omega)} \leq C$ for some constant, does it follow that $u \in L^\infty(0,T;L^\...
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0answers
24 views

Extending an approximation result in $\mathbb{L}^p$ for $p<\infty$ to $\mathbb{L}^\infty$?

studying a proof for the $p-$independence of the $UMD_p-$ property for Banach spaces there was the following technical lemma. Lemma: Let $1\leq p<\infty$ and $\epsilon >0$ be given. If $f$ is a ...
1
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1answer
45 views

Prove that there is a constant $ M $ such that $ \int|fg|dm\leq M \| f\|_{L^{p}} $ for all $ f\in L^{p}(\mathbf{R}) $.

I could not understand the last part of the proof of the following theorem: Let $ p\geq 1 $ and $ g $ be a measurable function such that $ \int|fg|dm<\infty $ for every $ f\in L^{p}(\mathbf{R}) $. ...
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0answers
40 views

If $ x \in \mathbb R^n$ with $\|x\|_\infty < 1$, how to make sense of this quantity $W(x):= \sum_{p=1}^\infty\|x\|_p^p$?

In some calculations of mine, I've stumbled on the following object, and I'm wondering if its a something recognisable. For $x \in \mathbb R^n$, let $\|x\|_\infty := \max_{1 \le j \le n}|x_j|$, and ...
2
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0answers
29 views

Convergence of difference quotient in $L^{p}(\mathbb{R}^{n})$

Let $f \in W^{1,p}(\mathbb{R}^{n})$, where $p \in (1,\infty)$. Let us define $f^{i}_{h}$ as $$ f^{i}_{h}(x) := \frac{f(x+he_{i}) - f(x)}{|h|}. $$ Prove that $$ ||f^{i}_{h} - D_{i}f||_{L^{p}(\mathbb{R}^...
1
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1answer
21 views

$L_p(\mu,X)$ is isometrically isomorphic to $\ell_p(X)$

In the book Banach Spaces of Vector-Valued Functions the authors present a demonstration for Proposition 1.6.4, pages 32-33: Let $1\le p \le + \infty$, if $(\Omega,\Sigma,\mu)$ is a $\sigma$-finite ...
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0answers
17 views

Norm of the resolvent and inequalities

I am considering the equation $$ \lambda q- \frac{\mathrm{d}^{2} q}{\mathrm{d} x^{2}}=g, $$ where $g\in L^p(X)$ and $q\in W^{2,p}(X)$. If, I know that $$ \mid \mid \lambda (\lambda I- \frac{\mathrm{d}^...
1
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1answer
52 views

$f_{n} \overset{\left\lVert .\right\rVert{.}_{1}}{\longmapsto} 0$ in $L^{1}([0,1])$ but not converging to $0$ almost everywhere.

I'd like to understand the costruction of the following function we should statisfy the requests of the title. We define $S_{n} := \sum\limits_{k=1}^{n}\frac{1}{k}$, with $a_{n} := S_{n} - \lfloor S_{...
1
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3answers
74 views

Compact integral operator on $L^\infty(\mathbb{R})$

Consider the following operator $\mathcal{L}: L^\infty(\mathbb{R})\rightarrow L^\infty(\mathbb{R})$: $$ \mathcal{L}v\equiv \int_{-\infty}^{\infty} K(x,y)v(y)dy. $$ Is this a compact operator if $K\in ...

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