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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1answer
33 views

How to prove Clarkson's inequality?

I do not know how to prove one of the four Clarkson's inequalities: let $u,v \in L^p(\Omega)$, if $1 < p < 2$, then $$ \bigg\lVert \frac{u+v}{2} \bigg\rVert_p^p + \bigg\lVert \frac{u-v}{2} ...
0
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0answers
13 views

Helmholtz decomposition of $v\in (L^2(\Omega))^3$

Let $\Omega\subset\mathbb{R}^3$ be a bounded domain with Lipschitz boundary $\partial\Omega$ and outward unit normal $n$. I want to study the characterize whether a vector function defined on ...
0
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0answers
14 views

Lp space to L-infinity space [duplicate]

Propositon: Let $(X,\mathscr{A},\mu)$ be a measure space with $\mu\lt\infty$ and let be $f\in L^{\infty}(X,\mathbb R; \mu)$. Then follows: $$\lim_{p\to\infty} \|f\|_{L^p} = ...
3
votes
1answer
60 views

$L^2(\mathbb R, \mu) $ a finite dimensional space.

Hi I find the following exercise. Honestly I'm not sure about my "answer", is incredible simple.t I don't know if make sense (in what part is necessary to use $L^2$?). I'd appreciate if someone can ...
1
vote
1answer
113 views

I want to prove $f\notin W^{1,1}(\mathbb{R},\gamma_{1})$

Let $\gamma_{1}=\mathscr{N}(0,I_{1})$ in $\mathbb{R}$ be the standard Gaussian measure. Consider the sequence $(f_{n})_{n\in\mathbb{N}}\in C_{b}^{1}(\mathbb{R})$ defined by $$f_{n}(x)=\begin{cases} ...
0
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1answer
35 views

On the reflexivity of the $L^p$-spaces

If $X$ is a normed vector-space, then $X^\ast$ is the normed vector-space of bounded linear functionals $X \rightarrow \mathbf{R}$. Assume $1 < p, q < + \infty$ such that $\frac{1}{p} + ...
2
votes
1answer
22 views

Inclusion of $L^r(\mu)$ in $L^q(\mu)+L^p(\mu)$

Let $(X,\mathcal{M},\mu)$ a positive measure space, then $L^r(\mu)\subset L^p(\mu)+L^q(\mu)$. How can I prove this? I know the standard inclusions for finite measure spaces, and spaces without ...
5
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0answers
105 views

Cyclic vector in $\ell^2(\mathbb{Z})$ space

Suppose we look at $\ell^2(\mathbb{Z})$, which contains vectors $c=(\dots,c_{-2},c_{-1},c_0,c_1,\dots)$ with $\sum|c(n)|^2<\infty$. Define the right-translation operator by $$R:\{c(n)\}\mapsto ...
2
votes
0answers
16 views

sum of uniformly bounded projections acting on a Lorentz sequence space

It is known that for every $k\in\mathbb{N}$ there is $N_k\in\mathbb{N}$ such that every $N_k$-dimensional subspace of $\ell_p$, $1<p<\infty$, contains uniformly complemented copies of $\ell_2^k$ ...
9
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0answers
169 views

$\int\limits_{\Omega}{uvdx}<\infty,\forall v\in H_0^1(\Omega)$ implies $u\in L^{6/5}(\Omega)$

Let $d=3$ and $\Omega\subset \mathbb R^d$ is a bounded Lipschitz domain and $u$ is a measurable function. A sufficient condition for the integral $\int\limits_{\Omega}{uvdx}<\infty,\forall v\in ...
1
vote
1answer
28 views

Show that if $1 \leq p < r < \infty$ then there is a finite constant $c$ such that $||u||_p \leq c||u||_r$ for every $u \in L_r(\mu)$

Let $(X, \Sigma, \mu)$ be a finite measure space. Show that if $1 \leq p < r < \infty$ then there is a finite constant $c$ such that $||u||_p \leq c||u||_r$ for every $u \in \ L^r(μ)$. Find a ...
3
votes
3answers
59 views

How “bounded” are $L^1$ functions?

I am well aware of the fact that $L^1-$functions are not necessarily essentially bounded. Take for instance the function $1/\sqrt{x}$ on $X=(0,1)$. However, can we say that they are "almost" bounded ...
0
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1answer
48 views

Relation between $L^1(\mathbb R)$ and $L^2(\mathbb R)$

I know that if $1\le p<q<\infty$ then $L^p\supset L^q $ and $l^p \supset l^q$. But what is the relation between $L^1(\mathbb R)$ and $L^2(\mathbb R)$? I guess there is no relation, i.e. ...
3
votes
2answers
48 views

Canonical injection from $L^p(0, 1)$ into $L^q(0, 1)$ continuous? How about compact?

Let $1 \le q \le p \le \infty$. I have two questions. Is the canonical injection from $L^p(0, 1)$ into $L^q(0, 1)$ continuous? Is the canonical injection from $L^p(0, 1)$ into $L^q(0, 1)$ compact?
2
votes
1answer
36 views

show that the function is in $L^r$

Let $f$ be a measurable function and $1 \le p < r < q < \infty$. If there is a constant $C$ such that $$\mu ( \{ x : |f(x ) | > \lambda \} ) \le \frac { C }{ \lambda ^p + \lambda ^ q} ...
4
votes
3answers
92 views

Subset of $\ell^2$ is precompact

Suppose we have a sequence of $a_i$ with some restrictions on it. Which restrictions must be to make set $$A= \left\{(x_i) \in \ell_2 \mid \sum\limits_{i\geqslant1} |a_i x_i|^2 \leqslant 1 \right\} ...
4
votes
1answer
40 views

$\ell^p \subset c_0$ with continuous injection.

Let $1 \le p < \infty$. How do I see that$$\ell^p \subset \left\{x ; \lim_{k \to \infty} x_k = 0\right\}$$with continuous injection?
5
votes
0answers
41 views

$T$ is a compact operator from $\ell^p$ into $\ell^p$ iff $\lambda_n \to 0$? [duplicate]

Let $E = \ell^p$ with $1 \le p \le \infty$. Let $(\lambda_n)$ be a bounded sequence in $\mathbb{R}$ and consider the operator $T \in \mathcal{L}(E)$ defined by$$T(x) = (\lambda_1x_1, \lambda_2x_2, ...
3
votes
1answer
50 views

If the sum of two independent random variables is $ L^{p} $, does it imply that each is $ L^{p} $?

Let $ X $ and $ Y $ be two independent random variables, i.e., $$ \forall a,b \in \Bbb{R}: \quad \textbf{Pr}(X < a,Y < b) = \textbf{Pr}(X < a) ~ \textbf{Pr}(Y < b). $$ Let $ p > 0 $ ...
7
votes
2answers
76 views

Prove a function is in $L^2[0,1]$

If $f\in L^2[0,1]$, and $$g(x)=\int_0^1\frac{f(t)\mathrm dt}{|x-t|^{1/2}},\quad x\in[0,1],$$ show that $\|g\|_2\le2\sqrt2\|f\|_2$. I tried Minkowski's integral inequality (with $p=1/2$, so ...
0
votes
3answers
97 views

The Schwartz space is dense in $L^p$

Is there any hint to prove that for every $1 \le p < \infty $ the Schwartz space is dense in $L^p$? Thanks so much.
2
votes
1answer
36 views

Does a function $f^p$ belong to $L^ {\infty}$ if $f \in L^{\infty}$ for $1 < p < \infty$?

I understand it should be so, considering the definition: $L^{\infty}( \Omega)= \{ f: \Omega \to \mathbb{R}\, \mid f$ is measurable and there is $C \in \mathbb{R^{+}}$ such that $|f(x)| \leq C$ ...
0
votes
1answer
46 views

Incompleteness of $\ell^1$ with respect to $\sup$ norm

I'm trying to make an example that shows $\ell^1$, that is the space of complex sequences that the sum of the norms of their components is finite, is not complete with respect to $\sup$ norm. And ...
5
votes
3answers
67 views

Is it possible to have $g\colon\Omega\to\Bbb C$ which defines an unbounded functional?

Let $\Omega$ be an infinite space with a nontrivial measure $\mu$. We define $L^p$ spaces as usual, then for $1<p<\infty$ if $\frac1p+\frac1q=1$, then $(L^p)^*=L^q$. This is all pretty much a ...
0
votes
1answer
27 views

Prove convergence to zero of $f(t + x) - f(x)$ in the $L^p$-norm [duplicate]

I'd like to show that $\Vert f(t + x) - f(x) \Vert_p \rightarrow 0$ as $t \rightarrow 0$, where $f \in L^p([0, \infty))$ and $\Vert \cdot \Vert_p$ is the usual norm on $L^p$. First I thought of using ...
0
votes
1answer
37 views

Can you proof this problem without using Riesz Representation Theorem?

I need to show that $c_0^* \cong l^1$ without using this theorem . I want to use the fact that if $Z = X \oplus Y$ then $Z^* = X^* \oplus Y^*$.
1
vote
1answer
38 views

Interpolation of a linear operator acting on a sequence of functions

Let $\mathbf{f} = \{f_{n}\}$ be a sequence of Schwarz functions and suppose $T$ is a linear operator which sends a given sequence of Schwarz functions to a given function in $L^{p}(\mathbb{R}^n)$ for ...
1
vote
1answer
42 views

interchange of $L^1$ and $L^{\infty}$ norm

Let $x,y \in \mathbb R^d,$ and $0\neq t \in \mathbb R.$ Define $f(y)= \sup_{x\in \mathbb R^d}\{e^{-\pi |y-tx|^2/ (1+t^2)} \}.$ My Question is: Is it true that $f\in L^{1}(\mathbb R^d)$? Is it ...
4
votes
1answer
42 views

The finite product of $L^p$ spaces is reflexive ($1<p<\infty$)

I am trying to understand the proof that the Sobolev Space $W^{1,p}$ is reflexive given in Functional Analysis, Sobolev Spaces and Partial Differential Equations, by Haim Brezis. There it is used ...
2
votes
1answer
56 views

Convergence of the $L^p$ norm to $L^{\infty}$ norm

Let $E \subset \mathbb{R}^n$ measurable. Prove that if there exist $p_0 \geq 1$ such that $f \in L^{p_o}(E) \cap L^{\infty}(E)$, then $f \in L^p(E)$ for all $p \geq p_0$ and $\|f\|_p \rightarrow ...
-1
votes
2answers
88 views

Dual space of $L^\infty$ is $L^1$ with the weak-* topology?

A friend of mine found a book in which the author said that the dual space of $L^\infty$ is $L^1$, of course not with the norm topology but with the weak-* topology. Does anyone know where I can find ...
1
vote
1answer
18 views

the Lp norm of the integral of a measurable function is bounded similar to Holders Condition

I have a final coming up in my Measure Theory class, and I found a question that I couldn't get a clean answer: show that for all $ f \in L^p[\mathbb{R}]$ there exists $C \in \mathbb{R}$ such that ...
1
vote
1answer
23 views

measurability of weak limit? or uniqueness of weak limit with sigma-algebras

I have a basic question about weak limits that I hope someone can clarify. Let $(\Omega,\mathcal{F},P)$ where $\Omega \subseteq \mathbb{R}^k$ be a probability space and let $\{f_n\}$ be a sequence of ...
0
votes
2answers
51 views

how to show its convergence in $Lp$ spaces?

Let $f_{n}\in L^{p}(\Bbb{R})(1\le p\le\infty)$, for $0<r<1$, if there exists a positive constant $C$,such that $||f_{n+1}-f_{n}||_{p}<Cr^{n}$, how to show $f_{n}$ converges in ...
0
votes
1answer
58 views

Are the three statements the same?

Let $\mathcal{S}(\mathbb{R}^n)$ be the Schwartz function on $\mathbb{R}^n$. Consider two statements which have the same proof. $$ f\in \mathcal{S}(\mathbb{R}^n)\,\,\Longrightarrow\,\,f\in ...
5
votes
1answer
48 views

Uniqueness in Riesz representation $L^p$-spaces.

We are in $L^p(\mu)$. Where $(\Omega, \mathcal{A},\mu)$ is an arbitrary measure-space. We have a bounded linear functional in this space $l$, by Riesz representation theorem we have that for $f\in ...
2
votes
0answers
28 views

Is the set of linear combinations dense in the set of the dual space of $l_p$?

Good day, Right now I'm working with the book "Functional Analysis" by Bachman and Narici, it is available on Google Books, see ...
2
votes
0answers
18 views

Fréchet differentiability of Nemyckij operator defined on $L^2$

I have been told the following. Suppose $\Omega\subseteq\mathbb{R}^n$ is a bounded borel set, $f$ is Carathéodory function on $\Omega\times\mathbb{R}=\{(x,s):x\in\Omega,s\in\mathbb{R}\}$, $f_s$, ...
3
votes
1answer
52 views

$L^\infty$ is complete - proof from exercises in Royden & Fitzpatrick's Real Analysis.

I've been reading up on some Analysis for my comp exams, and I couldn't find in my texts a proof of $L^\infty$ being Banach. Someone pointed me to the following exercise in Royden & Fitzpatrick. ...
1
vote
2answers
37 views

Form functions that are continuous at one point in L^\infty a Banach space.

Is the subspace $\{f \in L^\infty(\mathbb{R}) ~|~ f \text{ is continuous at } x=0 \}$ a Banach space? The norm is of course the essential supremum. Does the essential supremum even notice a single ...
1
vote
1answer
22 views

Convergence of second derivative in $L^2(\mathbb{R}^+)$

Could you find a sequence $f_n$ of smooth functions with compact support over the half line $\mathbb{R}^+$ such that $f_n$ converges in $L^2(\mathbb{R}^+)$ but such that the second derivatives ...
4
votes
1answer
34 views

Showing that the operator is bounded and find its norm.

I have this operator $T: L^p(0,\infty)\rightarrow L^p(0,\infty)$, $1<p<\infty$ : $(Tf)(x)=1/x\int_0^xf(t)dt$. I am supposed to show that it is bounded and fint its norm. I had an idea that ...
0
votes
1answer
47 views

Show that there is a measure $\mu$ and a subspace $Y$ of $L_p(\mu)$ such that $\bar{d}(X, Y) \leq \lambda$.

The following is Proposition $7.1$: Let $X$ be a Banach space, let $1 \leq p \leq \infty$ and let $\lambda \geq 1$. Assume that for every finite dimensional subspace $E$ of $X$ there is a subsapce ...
1
vote
1answer
31 views

Show that if $E$ is a finite dimensional subspace of $l_p$, there exists an integer $m$ such that $\| P_m(x) \| \geq (1 - \frac{1}{n}) \| x \|$

Show that if $E$ is a finite dimensional subspace of $l_p$, there exists an integer $m$ such that $$\| P_m(x) \| \geq (1 - \dfrac{1}{n}) \| x \|$$ where $x \in E$ and $P_m$ is a projection map from ...
1
vote
1answer
36 views

Fourier transforms having compact support

As we know, the fourier transform is a map $\mathcal{F}:L^1\rightarrow C_0$ (all with domain $\mathbb{R}$). Can one characterize the space of $f\in L^1$ such that $\mathcal{F}$ has compact support, ...
1
vote
0answers
24 views

Compute of two norms of a function of three variables

Let $f$ be a function defined on $\mathbb R^3$ by $$f(x,y,z)=\exp(-2\mathbb i\pi (x+y+z)) |x|^{1-k} |y|^{k-1} \operatorname{sign}(x) \operatorname{sign}(z),$$ where $sign(x)$ means the sign of $x$ and ...
3
votes
1answer
39 views

$L^2$ mapping is necessarily onto or not?

For $f \in L^2(\mathbb{R})$, let $$Tf(x) := \int_0^1 f(x+y)\,dy.$$Do we necessarily have that$$S: L^2(\mathbb{R}) \to L^2(\mathbb{R}),\text{ }Sf = f - Tf$$is onto?
2
votes
0answers
39 views

Disproving that a particular space is Banach

Let $E$ be a measurable set of finite measure and $1\leq p_1<p_2<\infty$. Consider the linear space $L^{p_2}(E)$ normed by $\|.\|_{p_1}$. Is this normed linear space a Banach space? My ...
0
votes
1answer
49 views

What values of $p$ give convergence to $0$ in $l^p$

Given a sequence $x_n \in l^p$ whose first $n^2$ members equal $\frac {1}{n}$, and all other entries $=0$, for what values of $p$ does the sequence converge to the zero sequence in $l^p$? So do I ...
1
vote
1answer
21 views

Embedding $\mathcal{l}^1 \subset \mathcal{l}^p$ continuous

Is the embedding of $\mathcal{l}^1$ in $\mathcal{l}^p$ for any $p\geq 1$ continuous?