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
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0answers
52 views

A question about general Marcinkiewicz interpolation theorem

The general Marcinkiewicz interpolation theorem states as following: If $T$ is a linear operator of weak type $(p_0,q_0)$ and of weak type $(p_1,q_1)$ where $q_0\neq q_1$, then for each $\theta\...
4
votes
1answer
62 views

Prove that $X_n \to X$ in $L^1$ if and only if $E(X_n1_{A}) \to E(X1_{A})$ uniformly on $A \in \mathcal{F}$

This is a probability exercise from the Karr's book called "Probability". Prove that $X_n \overset{L^1}{\rightarrow} X$ if and only if $$\sup_{A \in \mathcal{F}} \left|E(X_n1_{A}) - E(X1_{A})...
2
votes
1answer
56 views

What is the relation between $L^p([0,1])$ and the weak topology?

In my course of functional analysis, we always work with a Hausdorff locally convex space when we're creating the weak* and weak topology. My professor once stated that there is a reason for this ...
0
votes
1answer
51 views

Norm of pointwise product of Lp functions

Does the following inequality hold in $L_p$ spaces? $\|fg\|_p\leq\|f\|_p\|g\|_p$ How would I go about proving this? Do I need to apply Cauchy Schwarz?
8
votes
2answers
96 views

$(f_n)$ in $L^p(\Omega)$ satisfying $f_n(x) \to f(x)$ a.e. and $\|f_n\|_p \to \|f\|_p$, then $\|f_n - f\|_p \to 0$?

Let $1 < p < \infty$. If $(f_n)$ is a sequence in $L^p(\Omega)$ satisfying $f_n(x) \to f(x)$ a.e., $\|f_n\|_p \to \|f\|_p$, then does it follow that $\|f_n - f\|_p \to 0$? Edit. Here is my ...
1
vote
1answer
23 views

Difference between $C(\bar{D})$ and $L^\infty(D)$?

Let $D$ be an open and bounded set in $R^d$ with Lipschitz boundary. Denote by $C(\bar{D})$, the space of continuous functions $f : \bar{D} → R$. When equipped with the supremum norm $$\|f\|_{C(\bar{D}...
3
votes
1answer
41 views

$l^\infty$ equal to space of all sequence for which inner product with $l^1$ exists?

I know that the sequence space $l^\infty$ is equal to the dual of $l^1$ with respect to the $\|\cdot\|_1$ norm. But do we also have, that $$ l^\infty = \{f \in \mathbb R^{\mathbb N} \mid \sum_{k=1}^\...
0
votes
2answers
337 views

Find a sequence in $l^p$ but not in $l^q$, where $q < p$

I'm trying to find a sequence that is in $l^p$ but not in $l^q$, where $q < p$. Can anyone help?
1
vote
0answers
27 views

Approximation theory in Lp spaces (Reference Needed)

I am looking for some reference on approximation theory in Lp spaces. I have found a number of papers like: paper1 , paper2 etc. I was wondering if there is a book or a monograph that will contain ...
2
votes
0answers
56 views

approximations in lp spaces

If $f$ is a bounded measurable function, then, on every ball, the functions $f*\rho_{\epsilon}$ converge to $f$ in the mean and in measure. This corollary is from V.I.Bogachev, page 253 volume 1. I ...
0
votes
1answer
35 views

Convergence of Schwartz functions

I am proving or disproving the following statement: Let $f_n$ be a sequence of Schwartz functions in $\mathbb R^d$, such that $f_n$ converges to 0 uniformly. Is it true that $f_n$ converges to 0 in $L^...
3
votes
1answer
28 views

Proving that $x^{\alpha}(1+\Vert x\Vert^{2})^{-k}$ belongs to $L^{2}(\mathbb{R}^{n})$

Let $\alpha\in\mathbb{N}^{n}$ be a multi-index, i.e. $\alpha=(\alpha_{1},\dots,\alpha_{n})$ such that $x^{\alpha}:=\prod_{i=1}^{n}x^{\alpha_{i}}_{i}$. The modulus of a multi-index is defined as the ...
1
vote
1answer
41 views

Lp spaces (Hölder, Minkovski)

Let $1<p<\infty$ and $f\in L_p(0,\infty)$. Show that $$\lim_{x\to\infty}\frac{1}{x^{1-\frac{1}{p}}}\int_0^x f(t)dt=0$$ by assuming that f is compactly supported. Any idea so it can help me how ...
0
votes
1answer
22 views

Is a $C^2$ representative unique in $L^2([0,1])$?

I am reading a paper that deals with the solution for the Sturm-Liouville problem's uniqueness of solutions by defining an operator $K$: $$K:L^2([0,1]) \to L^2([0,1])$$ $$f \hspace{0.5cm} \...
1
vote
3answers
67 views

Showing that sequences such that $\sum_{n=1}^\infty {x_n\over n} =1$ form a closed subset of $l^2$

Let $H= \left\{(x_n)\in\ell^2 : \sum_{n=1}^\infty {x_n\over n} =1\right\}$ I need to show that H is closed in $l^2$. then, it is sufficient to show that, closure of H $\subseteq$ H. let x=($x_i$)$\...
0
votes
1answer
67 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} \bigg\...
1
vote
0answers
25 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 $\Omega$...
0
votes
0answers
16 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: $$\lim_{p\to\infty} \|f\|_{L^p} = \|f\|_{L^{\infty}}.$$...
3
votes
1answer
65 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
121 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,...
0
votes
1answer
45 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} + \frac{1}{...
2
votes
1answer
25 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
votes
0answers
353 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 \{c(...
2
votes
0answers
18 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
votes
0answers
227 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 H_0^...
1
vote
1answer
29 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
65 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
votes
1answer
50 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. $L^1(\...
3
votes
2answers
58 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
38 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
99 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
55 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
44 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
53 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
81 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 the ...
1
vote
3answers
191 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.
3
votes
1answer
41 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
47 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
95 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
32 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
40 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
42 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
52 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
95 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 \|f\|...
-1
votes
2answers
164 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
44 views

Haar functions form a complete orthonormal system

I want to show that the Haar functions in $L^2([0,1])$ forms an orthonormal basis: Let $$f = 1_{[0, 1/2)} - 1_{[1/2,0)} \ \ \mbox{,} \ \ f_{j,k}(t) = 2^{j/2}f(2^jt - k).$$ Let $\mathscr{A} = \{(j.k) :...
1
vote
1answer
21 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
30 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 ...
-2
votes
2answers
66 views

How to show the convergence in $L^p$ spaces? [closed]

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 $L^{p}(\Bbb{...