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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3answers
52 views

An example of a function in $L^1[0,1]$ which is not in $L^p[0,1]$ for any $p>1$

Title says most of it. Could you help me find an example? It is easy obviously to show a function that would not be in $L^p[0,1]$ for a specific $p$ (say $(1/x)^{1/p}$, but I can't see how it would ...
2
votes
1answer
21 views

Are Schwartz functions in $L^{p}$ for $0 < p < 1$?

Let $S(\mathbb{R}^{d})$ denote the Schwartz functions in $\mathbb{R}^{d}$. I know that $S(\mathbb{R}^{d}) \subset L^{p}(\mathbb{R}^{d})$ for $1 \leq p < \infty$. Is $S(\mathbb{R}^{d}) \subset ...
0
votes
0answers
11 views

$L^p$ is a quasi normed space for $0<p<1$ [duplicate]

I know that $L^p$ is a vector space for $p>0$ and a normed space for $p \geqslant 1$ now I need show that for $ 0<p<1$ and $f,g \in L^p$ exist $K \in \mathbb{R}$ such that $||f+g||_p ...
3
votes
1answer
23 views

$L^\infty(S^1)$ is not separable

Let $S^1$ be the unit circle and $L^\infty(S^1)$ the space of measurable functions $f:S^1\to\mathbb{C}$ such that $\|f\|_\infty<\infty$. (In fact $L^\infty(S^1)$ consists of equivalence classes of ...
0
votes
2answers
25 views

Interpolation inequality in sobolev space

Let $U$ be a bounded, connected open subset of $\mathbb R^n$ with $C^1$ boundary $\partial U$. Asume $|\beta| \leq k-1$ and $k$ is a integer. Show that for each $\epsilon >0$ there exists a ...
2
votes
1answer
48 views

$f \in L^1$, but $f \not\in L^p$ for all $p > 1$

"Find an $f \in [0,1]$ such that $f \in L^1$ but $f \not\in L^p$ for any $p > 1$." I've thought about doing something like $$f(x) = \frac{1}{x}$$ where $|f|^p = \frac{1}{x^p}$ doesn't converge ...
0
votes
0answers
15 views

Orthonormal bases in L^p

Let $(X,\mu), (Y,\nu)$ be $\sigma$-finite measure spaces. Suppose $\phi_1$ and $\phi_2$ are orthonormal bases for $L^p(\mu)$ and $L^p(\nu)$, respectively. How do I show that $\{f\otimes g: f\in ...
0
votes
0answers
18 views

Product Measure in $\mathbb{L}^p$

Let $(X,\mathscr{A},\mu_1)$ and $(Y,\mathscr{B},\mu_2)$ be $\sigma$-finite measure spaces and let $\nu$ be the product measure. Let $p\in[1,\infty)$, $f\in L^p(\mu_1), g\in L^p(\mu_2)$, and define $(f ...
0
votes
1answer
33 views

Is $L^\infty$ compactly embedded in $L^1$?

I'm trying to find a contraction example to show that the space $L^\infty$ is not compactly embedded in $L^1$ with the Lebesgue measure. Please help me!
1
vote
0answers
37 views

Question about Young's Inequality

I came across Young's Inequality for convolutions, stated as: Let $f\in L^p, g\in L^q$, where $p,q,r \in [1,\infty]$ and $p^{-1}+q^{-1}=1+r^{-1}$. Then $f*g$ is defined $m$-a.e. on $\mathbb{R}^d$, ...
1
vote
0answers
28 views

Equality condition for convolution's $L^p$ norm.

Suppose that $1< p< \infty$, $f\in L^1(R)$, and $g\in L^p(R)$ and that $\|f*g\|_p=\|f\|_1\|g\|_p$. Show that then either $f=0$ a.e or $g=0$ a.e I have solved for $g=0$ a.e. if $||f||_1>0$ ...
5
votes
0answers
63 views

Solving a functional equation in $L_2(\mathbb{R})$

Let $e\left(x\right)=e^{2\pi ix}$ and let $F$ be an arbitrary complex-valued function in $L^2 (\mathbb R)$. I am trying to solve the following functional equation (or rather family of equations): ...
3
votes
0answers
19 views

Simple $L^p$ space question [duplicate]

Let $X$ be a measure space with positive measure $\mu$. If $\|f_n-f\|_p\to 0$ (i.e. $f_n$ converges to $f$ in the $L^p(\mu)$ norm) does it follow that $f_n(x)\to f(x)$ almost everywhere?
1
vote
1answer
64 views

Proving this set is dense in $\ell^2$

I found this weirdest question and was wondering how could this be proved. This question is a part of a beautiful semi-constructive built of two dense disjoint convex sets in $\ell^2$, which I find ...
1
vote
0answers
16 views

Equivalence of condition on function

In a book I am studying it states that a condition on a function $g$ as follows: Given the function $g: \Omega \times \mathbb{R} \mapsto \mathbb{R}$ is a Caratheodory function satisfying $$\sup_{|u| ...
5
votes
1answer
81 views

orthonormal sequence in $L^2[0,1]$ - how to prove these following equivalent terms?

I've been asked this following very interesting question and would like some help figuring out why it is true :) Let $u_n$ be an orthonormal sequence in $L^2[0,1]$ Prove that the following are ...
1
vote
1answer
17 views

what does support of convolution of functions says geometrically?

Let $f,g \in L^{1}(\mathbb R)$ we define $f\ast g(x)= \int_{\mathbb R} f(x-y)g(y) dy $ for all most all $x,$ and denote $\text{supp} (f)$ the support of $f.$ Fact: If $A$ is the closure of $\{x+y: ...
4
votes
2answers
72 views

$e_n \to 0$ weakly in $l^\infty$

Given the the sequence $(e_n)_n$ in $l^\infty$, I want to show that that $e_n$ converges weakly to $0$ in $l^\infty$, i.e. $$e_n\rightharpoonup 0 \text{ as } n\to \infty.$$ By $e_n\in l^\infty$, ...
1
vote
1answer
23 views

Hardy-Littlewood maximal function $Mf$ is greater than $f$

Suppose that $f\in L^{1}(\mathbb R)$ and $x\in \mathbb R.$ We denote $B_{x}$ by the ball in $\mathbb R$ with centred $x,$ and $|B_{x}|=$ length(Lebsgue measure) of $B_{x}.$ Put, $Mf(x)=\sup ...
0
votes
0answers
18 views

Basic question about $L_1$ integrable functions [duplicate]

I need to show that if a function is in $L_1$, i.e., the integral of $|f|$ over the real line (Lesbesgue measure) is finite, then the limit as $n$ goes to infinity of the integral of $f$ (from $n$ to ...
3
votes
1answer
27 views

A function in $L^p(E)$ for only $2 \leq p < 3$

Give a set $E$ and a function $f$ on $E$ such that $\{p:f \in L^p(E)\} =[2,3)$. This was a homework problem (whose due date has passed) that my brightest fellow classmates AND the class TA ...
-2
votes
1answer
85 views

Determine if these two norms are equivalent

Let we have the space $C[a,b]$ (the space of all functions that are continuous on closed interval $[a,b]$). And we have two norms on this space: $$\|X\|_1= \max_{t\in [a,b]} | x(t) |$$ $$\|X\|_2 = ...
3
votes
1answer
22 views

Is the $\sigma$-finiteness condition necessary to ensure that $L^p(\mu)$ is reflexive?

Suppose $(X,\mu)$ is a measure space, and $p,q>0$ such that $1/p+1/q=1$. We know that if $\mu$ is $\sigma$-finite, then $L^p(\mu)$, $L^q(\mu)$ are reflexive and dual to each other. The proof could ...
1
vote
0answers
47 views

Proof Riesz Representation Theorem (bounded linear functional in Lp)

I have a little problem with this proof (I'm using Royden), can you help me? Let $F$ be a bounded linear functional on $L^p$, $1 \leqslant p \leqslant \infty$. Then there is a function $ge \in L^q$ ...
1
vote
1answer
44 views

$L^\infty$, $L^p$ question

Suppose $f \in L^0$. I read that for a general measure space, if $\mu(X)<\infty$, then we cannot have that both $||f||_\infty< \infty$ and $||f||_p=\infty$ for every $p\in (0,\infty)$, but if ...
3
votes
1answer
50 views

Weak convergence in $l_p$ implies pointwise convergence?

Could someone please share their thoughts on this one: Consider at $l_p(Y)$, for $1<p<\infty$ with the counting measure on $Y$. Show that if a sequence weakly converges in $l_p(Y)$ then it ...
1
vote
2answers
20 views

Translation operator and continuity

I came across a text that proves that translation operator $T_a(f):=f(x-a)$ where $a\in\mathbb{R}^n$ and $f\in L^p(\mathbb{R}^n)$ is continuous. The proof follows: ...
1
vote
1answer
21 views

Formulation of Fourier transform

I would like to know about Fourier transform more. I attended a standard lecture of mathematics, but we did not talk about Fourier transform on $L^2$ much, nor the theory of $L^2$. We only defined it ...
1
vote
1answer
40 views

How to prove that the function $\tan(x)1_{(0,\pi/2)}$ lies in $L^p$ for $p\in (0,1)$?

I am attempting to prove that the function $\tan(x)1_{(0,\pi/2)}$ lies in $L^p$ for $p\in (0,1)$. To do this, I want to calculate the integral $\int_{0}^{\pi/2}\tan(x)^{p}\,dx$ for various $p$. ...
0
votes
1answer
21 views

An inequality for functions in $L^1$ and $L^2$.

Given $f\in L^1(X)$, show that for every $\rho>0$ one has: $\mu({|f|>\rho})\leq \rho^{-1}\int|f|d\mu$. I think in order to prove this, we use the fact that $\int|f|d\mu= \lim_{\rho\to ...
1
vote
1answer
38 views

Weak convergence in $\mathcal{l}_p$ and coordinatewise convergence

Let $x^n=(x^n_1, x^n_2,...)$ be a bounded sequence in $\mathcal{l}_p$ for $1<p<\infty$ and such that $x^n_i$ converges to $x_i$ for all $i\in\mathbb{N}$. I'm trying to prove that ...
1
vote
0answers
29 views

Does an inequality between kernels imply an inequality between the norms of integral operators?

Assume that $g(x,y)$ and $h(x,y)$ are two positive functions such that $0<g<h$ and assume that $$T_g, T_h : L^2(B^n,R)\to L^2(B^n,R)$$ are integral operators defined by $$T_k[f](x)=\int_{B^n} ...
2
votes
1answer
52 views

Help in a problem about Lebesgue integration inequality

Let $ (X,\mathcal{S},\mu)$ be a finite measure space, let $f$ be $\mathcal{S}$-measurable and let $E_{n}:= \{x\in X :n-1\le |f(x)|<n\}$ for $n=1,2,\dots$ Show that: $$f \in ...
1
vote
2answers
28 views

A question about inclusion of $L^r(\mu)$ spaces for different $r$ and different measures $\mu$

For some measures, the relation $r<s$ implies $L^r(\mu)\subseteq L^s(\mu)$ ; for others, the inclusion is reversed; and there are some for which $L^r(\mu)$ does not contain $L^s(\mu)$ if $r\ne ...
1
vote
0answers
21 views

Discontinuous functionals on $L^p$

Using the axiom of choice and a Hamel basis for a normed space, one can prove the existence of everywhere defined discontinuous linear functionals. My question: Does there exist a discontinuous ...
0
votes
2answers
11 views

$\int_{\mathbb R} \sup_{|y|>|x|} \frac{1}{(1+|y|)^{n}} dx < \infty$ for large $n$?

It is true that $\int_{\mathbb R} \frac{1}{(1+|y|)^{n}} dy <\infty$ for large $n\in \mathbb N.$ My Question is: Can we expect $\int_{\mathbb R} \sup_{|y|>|x|} \frac{1}{(1+|y|)^{n}} dx < ...
0
votes
1answer
35 views

Weak convergence on L^p

Let Let $X=[0,1]$ with the Lebesgue measure, find a sequence $\{f_n\}$ of measurable functions $f_n:X \rightarrow{ \mathbb{R} } $ such that: $f_n(x)\rightarrow{0}$ almost everywhere $x∈[0,1]$ ...
2
votes
0answers
17 views

Relation between a.s. convergence and weak convergence in in $L^p$ spaces

Is there a relation between almost-sure convergence and weak convergence of $f_n\in L^p(\mathbb{R})$? (i.e. convergence if tested with $L^{p'}$) I know that none implies the other (see masses ...
0
votes
1answer
13 views

Is contraction of $L^{2}$ function is again $L^{2}$ function?

Suppose that $f\in L^{2}(\mathbb R).$ Let $g:\mathbb R \to \mathbb C$ be a contraction of $f$, that is, $|g(x)-g(y)|\leq |f(x)-f(y)|$ for all $x, y\in \mathbb R.$ My Question is: Is it true that ...
3
votes
1answer
23 views

Positivelinear operator on $L^p$-spaces

Suppose $1<p<\infty$. A linear operator $T \colon L^p(\Omega)\to L^p(\Omega)$ is positive if $f \geq 0$ imply $T(f)\geq 0$ (where $\Omega$ is a measure space). 1) Does there exist a positive ...
3
votes
3answers
84 views

How to prove that $L^p [0,1]$ isn't induced by an inner product? for $p\neq 2$

I'd like to know how could i prove that $L^p [0,1]$ isn't induced by an inner product? (For $p\neq 2$, including $p=\inf$). It is clear to me that i would need to find two functions $f$, $g$ in $L^p$ ...
1
vote
1answer
31 views

L_p spacece and subset of them

let $(\Omega ,A,\mu)$ be a measure space and let $1<p<q<r<\infty$. show that $$L^q(\Omega)\subset L^p(\Omega)+L^r(\Omega)$$ answer:Iknow that I can assume that $f\in ...
3
votes
1answer
38 views

Different norm on $\ell_p$-space and Hilbert space

We define $\ell_p=\{(x_n)_{n\in{\mathbb{N}}}\in\mathbb{C}^\infty:\sum_n{|x_n|^p}<\infty\}$. With the usual usual norm $||.||_p$ this becomes a Bancach space. Also we have the usual inner product : ...
1
vote
1answer
48 views

Does uniform convergence imply $L^1$-convergence?

Suppose that $(X,M,\mu)$ is a measure space. Is it true that if $\,f_n \in L^1(X)\,$ and $\,f_n\,$ converges uniformly to $f$, then $\,f \in L^1(X)\,$ and $\,\int_X f_n\,d\mu\,$ converges to ...
1
vote
2answers
71 views

Is $\ell^1$ an inner product space?

Considering the parallel result for $L^p$ spaces, I would guess that $\ell^1$ is not an inner product space. The proof would presumably follow by providing counterexample sequences $x = (x_n) $ and ...
3
votes
2answers
43 views

Why is convergence w.r.t $\mathcal L^p$-norm of a sequence $(f_n)$ of $\mathcal E-\mathcal B(\mathbb R)$-functions called “convergence in $p$-mean”?

In measure-theory, why is convergence with respect to the $\mathcal L^p$-norm of a sequence $(f_n)_{n \in \mathbb N}$ of $\mathcal E-\mathcal B(\mathbb R)$-measurable functions called "convergence in ...
1
vote
1answer
24 views

Finite parameter integral implies finite norm

Need a bit of help with a parameter integral problem. We have, $X$ is a finite measure space with measure $\mu$ and $f:X\rightarrow [0 , \infty)$ is a measurable function. The parameter integral ...
3
votes
1answer
31 views

Proving functions are in $L_1(\mu)$.

Let $(\mathcal{X}, \mathcal{A}, \mu)$ be a measure space. Take $f,g \in L^1(\mu)$. Prove that $\sqrt{f^2+g^2}$ and $\sqrt{\vert fg\vert}$ are in $L^1(\mu)$. First, I prove that $h = ...
0
votes
1answer
25 views

Proof completeness of $L^p$

I'd like to check if I understood the proof that $L^p$ is complete ($1 \le p <+\infty$). I have to use the following fact: in a metric space, if a Cauchy sequence has a convergent subsequence then ...
1
vote
1answer
26 views

Embedding of Lp spaces

I've managed to prove that for $ 1\leq p < q \leq +\infty $ we have an inclusion (embedding) $ L_q([0,1],\lambda) \rightarrow L_p([0,1], \lambda) ~~ (\lambda $ being Lebesgue measure). The trouble ...