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.

1
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2answers
112 views

Examples that are not Lebesgue integrable for any $p$

I've been trying to think up different examples of functions such that $EZ^p = \infty$ (with $Z>0$) for all $p$, but each time it becomes rather messy. Can anyone suggest some interesting but ...
5
votes
1answer
145 views

Limit problem for $L^p$ function

I am having problems with proving the following: Let $f$ be a $L^p$ function on $[0,1]$, $f:[0,1] \to \overline{\mathbb{R}}$. Prove that $$\lim_{t \to \infty} t^p \mu(x: |f(x)| \geq t) = 0.$$ ...
14
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1answer
2k views

How to show convolution of an $L^p$ function and a Schwartz function is a Schwartz function

We have the Schwartz space $\mathcal{S}$ of $C^\infty(\mathbb{R^n})$ functions $h$ such that $(1+|x|^m)|\partial^\alpha h(x)|$ is bounded for all $m \in \mathbb{N_0}$ and all multi-indices $\alpha$. ...
3
votes
2answers
385 views

Question from Folland, criteron for a function to belong to $L^p$

This question is from Folland 6.38, Show that $f \in L^p $ iff $\sum_{k=-\infty}^ {\infty} 2^{pk} \mu \{{x: |f(x)|>2^{k}}\} \lt \infty$ If $f \in L^p $, I applied the Chebyshev's inequality But ...
1
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2answers
255 views

Is the inclusion map of $\ell^1(N)$ in $\ell^2(N)$ bounded and dense?

I am looking for an idea to prove if the inclusion map from $\ell^1(N)$ to $\ell^2(N)$ bounded and does it have a dense image. And why is the set $A:=\{x: ||x||_1\le 1\} \subset \ell^2(N)$ closed and ...
2
votes
1answer
277 views

Closed subspace of $L^1[0,1]$

The statement I need to prove is following. Let $S$ be a closed subspace of Lebesgue space $L^1[0,1].$ Assume that for every $f\in S$ there exists a number $p(f)>1$ such that $f\in L^{p(f)}[0,1].$ ...
1
vote
1answer
63 views

Limit and Integral sign in $L^2$.

If $\lim_{k \to \infty} \| u_k - u \|_{L^2(\Bbb R^n)} = 0$ then how can I show that $$ \lim_{k \to \infty} \int_{\Bbb R^n} u_k v = \int_{\Bbb R^n} uv$$ for any $v \in L^2 (\Bbb R^n)$?
6
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1answer
3k views

Liapunov's Inequality for $L_p$ spaces

Let $1 \leq p,q < \infty$ and $0 \leq \lambda \leq 1$. If $r = \lambda p + (1 - \lambda)q$ and $f \in L_p \cap L_q $, then $$||f||_r^r \leq ||f||_p^{\lambda p} ||f||_q^{(1 - \lambda)q} \tag{*}$$ ...
5
votes
2answers
735 views

When multiplication operator is an bounded below/ open mapping/isometry/quotient map?

Let $(\Omega,\Sigma)$ be a $\sigma$-finite measurable space. Let $\mu,\nu\in \mathcal{M}(\Omega)$ be $\sigma$-additive measures on $\Omega$. Assume we are given $p,q\in[1,+\infty]$ and a measurable ...
3
votes
1answer
796 views

Rademacher functions form an orthonormal system but not an orthonormal basis

I would like to know how to show that the functions $$r_n(t)=\operatorname{sgn}\big(\sin(2^n \pi t)\big)$$ (where $\operatorname{sgn}$ is the sign function) form an orthonormal system but not an ...
4
votes
1answer
456 views

$L^2$ norm inequality

I need some help with this homework question. I was asked to provide an example of a $n$-dimensional subspace $W$ of $L^2[0,1]$ such that all functions in that subspace with $L^2$ norm equal to $1$ ...
4
votes
1answer
290 views

How is this book applying Fubini/Tonelli without assuming $\sigma$-finiteness?

I am reading about $L^p$ spaces on this google book and in proposition 1.1.4 (page 4) it writes $$ p\int_0^\infty \alpha^{p-1} \int_X \chi_{\{x:|f(x)|>\alpha\}}d\mu(x)d\alpha = \int_X \int_0^{|f(x)|...
2
votes
1answer
123 views

Does $\Vert f \Vert_p = \sup_{\Vert g \Vert_q=1}\int fg d\mu$ fail if $f \notin L^p$?

I know that for $p \in [1,\infty]$ if $X$ is $\sigma$-finite (for the $p=\infty$ case) we have $$ \Vert f \Vert_p = \sup_{\substack{g \in L^q\\\Vert g \Vert = 1}} \int_X fg d\mu. $$ I always see it ...
0
votes
1answer
127 views

Convergence in $L^2$ space.

Let $u_k , u \in L^2 (\Bbb R^n)$ for $k \in \Bbb N$. Assume that $f : \Bbb R^n \to \Bbb R$ is continuous and $|f (u_k) | \leqslant M$ , $|f(u) | \leqslant M$ for some $M >0$. If $u_k$ converges to $...
5
votes
2answers
2k views

$L^p$ norm of multivariate standard normal random variable

Given $X_i\sim \mathcal{N}(0,1)$ what is the behaviour of $$ ||X||_{l^p}=(\sum_{i=1}^n|X_i|^p )^{1/p}$$ as $n\rightarrow \infty$? For $p=2$ results about $\chi$-distribution tell us that $$\mathbb{P}...
3
votes
1answer
138 views

Negative integral on intervals implies negative function?

Let $f \in L^1([0,1])$ be such that for all $t \geq s$, $\displaystyle \int_s^t f(u)du \leq 0$. Is it true that $f\leq 0$ almost everywhere?
2
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1answer
322 views

weak vs. norm compactness in $\ell_1$

So I'm trying to show that weakly compact sets in $\ell_1$ are norm-compact. I've already proven that weak sequential convergence implies norm convergence. I think the idea I want to go with is to ...
1
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1answer
179 views

Show a function is in $L_\infty$

Let's assume we're working on a measure space $(X,\Sigma,\mu)$, where $\mu$ is a $\sigma$-finite measure. Suppose that $g$ is a measurable function such that $\forall f\in L^2$, $||fg||_2\leq ||f||...
3
votes
2answers
265 views

In $\ell^1$ but not in $\ell^2$?

Can a sequence $f:\mathbb{Z}\to\mathbb{C}$ be in $\ell^1$ but not $\ell^2$? (any one counter example will suffice)
7
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2answers
170 views

On $L^p$ and $\ell^p$

If a continuous and infinitely differentiable function $f(x): \mathbb{R}\to\mathbb{C}$ is in $L^p$, is it also true that $f(n),\ n\in \mathbb{Z}$ is in $\ell^p$?
1
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1answer
316 views

prove a subset of squence space $l^p$ closed in strong topology

Let $l^p$ be the space of $p$-summable sequences. von Neumann constructed a subset of $l^p$ space $$S=\{X_{mn}: m,n≥1\}$$ where $X_{mn}\in l^p$ are defined by $X_{mn}(m)=1, X_{mn}(n)=m$ and $X_{mn}(...
2
votes
1answer
161 views

Isometric embeddings of $\ell_q^m$ into $\ell_p$ and $L_p$ for $p,q\in[1,+\infty]$

I'm looking for articles describing (or proving nonexistence) of isometric embeddings of $m$-dimensional space $\ell_q^m$ into $L_p$ and $\ell_p$ for $q,p\in[1,+\infty]$. Since $\ell_q^m$ is finite ...
0
votes
1answer
422 views

Dual space of the function $f$ in Fourier Transform

Let $f\in L^1{(\mathbb{R})}$. Why the Fourier Transform $\hat{f}\in L^{\infty}{(\mathbb{R})}$. Is it because $(L^1{(\mathbb{R})})^*=L^{\infty}{(\mathbb{R})}$?
3
votes
3answers
285 views

$L_p$ Spaces and limits of translated functions

If $g\in L^p(\mathbb{R}^n)$ and $1\leq p<\infty$ then show $$\lim_{|t|\to \infty}\lVert g_{(t)}+g\rVert_p=2^{1/p}\lVert g\rVert_p,$$ where $g_{(t)}(x):=g(t+x)$. Any hints? Try to give me only ...
8
votes
2answers
507 views

Is the injection $\ell^p \subset \ell^q$ continuous for $p<q$?

It is easy to show that $\ell^p \subset \ell^q$ when $1 \leq p<q \leq + \infty$, but is the injection continuous? If so is $\ell^{\infty}$ the direct limit $\lim\limits_{\rightarrow} \ \ell^p$ as ...
1
vote
2answers
100 views

$f_n$ $\in$ $L_2(\mu)$, the limit $ f \in L_2(\mu)$

If $f_n \in L_2(\mu)$, $f_n\rightarrow f$ almost everywhere, this is not enough to conclude $f\in L_1(\mu)$. But is it enough to conclude whether $f\in L_2(\mu)$ or $$\lim_{n \to \infty}\int_{R}{|f_n(...
1
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0answers
82 views

Under what condition two Lp spaces contain the same functions [duplicate]

Possible Duplicate: When $L_p = L_q$? Can anyone tell me under what condition these two spaces $L^p(\mu)$ and $L^s(\mu)$ contain the same functions?
1
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1answer
274 views

How to bound $L^p$ norm of a product

I am trying to show that if I can approximate two characteristic functions $\chi_A,\chi_B$ by simple functions involving only a particular set of characteristic functions, then I can approximate $\...
0
votes
1answer
127 views

what is the closure of $\mathbb{Q}^\mathbb{N}$ in $l^\infty$?

I was wondering that since $l^\infty$ is not separable, which means that there is not a countable dense set in it. However the set $\mathbb{Q}^\mathbb{N}$ is countable (am I right in this?). So what ...
2
votes
2answers
71 views

Convergence of a sequence in $L^1(\mathbb{R}^3)$

All function spaces are over $\mathbb{R}^3$. Let $u_n \in C^\infty_0$, $u_n\rightarrow u$ in $L^1$. Let $v\in L^1_\text{loc}$ be such that $uv \in L^1$. Does $u_n v \rightarrow uv$ in $L^1$? What ...
6
votes
2answers
1k views

Proof of Pitt's theorem

I'm reading the book Topics in Banach Space Theory by Albiac F. Kalton N. J. I got stuck at the proof of Pitt's theorem. In the second paragraphs authors tries to prove ad absurdum that for weakly ...
4
votes
2answers
3k views

$\ell_{p}$ space is not Hilbert for any norm if $p\neq 2$

My question is motivated by this one: $\ell_p$ is Hilbert space if and only if $p=2$ Maybe it is a simple thing or im just confused but, suppose we are given any norm in $\ell_{p}$ for $p\neq 2$. ...
20
votes
1answer
10k views

$\ell_p$ is Hilbert if and only if $p=2$

Can anybody please help me to prove this: Let $p$ be greater than or equal to $1$. Show that for the space $\ell_p=\{(u_n):\sum_{n=1}^\infty |u_n|^p<\infty\}$ of all $p$-summable sequences ...
3
votes
1answer
362 views

Closure of $l_1$ in $l_\infty$

Suppose we have a set $A$ which is the set of all sequences that satisfy $|x_n|\xrightarrow{} 0$. If we consider $l_1$ to be a subset of $l_\infty$. Show that the closure of $l_1$ in $l_\infty$ equals ...
0
votes
1answer
208 views

The distance between a point and a set

It is a problem in my homework. Let $$ X = \{x \in C[0,1] : x(0) = 0\} $$ with norm $\Vert\cdot\Vert_\infty$. Denote $$ M =\left\{ x \in X : \int\limits_0^1 x(t)=0\right\} $$ If $\Vert x_0\Vert_\...
8
votes
1answer
681 views

Isometry between $L_\infty$ and $\ell_\infty$

It is known that there exist some isomorphism between $L_\infty$ and $\ell_\infty$, which is not explicit at all. Could someone tell me whether there exist an isometric isomorphism between $L_\infty$...
15
votes
1answer
2k views

A Hamel basis for $\ell^p$?

I am looking for an explicit example for a Hamel basis for $\ell^{p}$?. As we know that for a Banach space a Hamel basis has either finite or uncountably infinite cardinality and for such a basis one ...
14
votes
2answers
2k views

Characterization of weak convergence in $\ell_\infty$

Is there some simple characterization of weak convergence of sequences in the space $\ell_\infty$? If yes, is there some similar claim for nets? I was only able to come up with a characterization of ...
8
votes
2answers
2k views

How to prove that the $L^p$ spaces are infinite dimensional

It is well-known that (given a measure space $(S,\mathcal A,\mu)$ and $1\le p\le\infty$) the Banach space $L^p(S,\mathcal A,\mu)$ has infinite dimension. Is there an easy way to proof this statement (...
3
votes
1answer
675 views

Boundedness of multiplication operator on $L^p$ spaces.

I am asking myself when a $L^p\to L^q$ multiplication operator is continuous. The following should be true: Let $a:[0,1]\to\mathbb{C}$ be a measurable function. Let $T_a: L^p([0,1])\to L^q([0,1])$, ...
4
votes
3answers
729 views

Every absolutely continuous function with integrable derivative tends to zero at infinity

I am given $f,f' \in L^1(\mathbb{R})$, and f is absolute continuous, I want to show that: $$\lim_{|x|\rightarrow \infty} f(x)= 0$$ Not sure how to show this, I know that $f(x)=\int_0^x f'(t) \, dt+f(...
8
votes
1answer
4k views

Inequalities in $l_p$ norm

I'm having difficulty with the following problem. Any help would be appreciated. Problem: Consider the sequence spaces $l_p$ with the usual norm. If $1\le p\le q\le \infty$, I want to show the ...
2
votes
2answers
2k views

Completeness of $\ell^2$ space

I was reading up and it says that $(\ell^2,\|.\|_2)$ is complete. I know that a metric space $X$ in which every Cauchy sequence converges to an element of $X$ is called complete. And I know that a ...
12
votes
1answer
5k views

Fourier transform in $L^p$

Let the $f$ be a function in $L^s$ where $s \in [1,\infty) $. For which $r$ Fourier transform $\hat{f}$ belongs to $L^r$? I'd be grateful for any kind of help including providing a literature or ...
4
votes
1answer
4k views

Using Lusin's Theorem to show that continuous functions are dense in $L^p$

Lusin's theorem says that in a finite measure space, given a measurable function $\varphi$, for every $\varepsilon \gt 0$ there exists a continuous function $g$ such that $$ \mu\left(\{x : \varphi(x)\...
12
votes
2answers
3k views

What is the predual of $L^1$

Is there a nice characterization of the predual of $L^1$? So, what does the space $X$ look like, such that $X^*=L^1$, where the star denotes the dual of a Banach space. How do you start to find such ...
4
votes
2answers
4k views

Convergence in $L^{\infty}$ norm implies convergence in $L^1$ norm

Let $\{f_n\}_{n\in \mathbb{N}}$ be a sequence of measurable functions on a measure space and $f$ measurable. Assume the measure space $X$ has finite measure. If $f_n$ converges to $f$ in $L^{\infty}$-...
7
votes
2answers
3k views

Cesàro operator is bounded for $1<p<\infty$

The Cesàro operator $T\colon \ell_{p}\to\ell_{p}$ is defined by $(Tx)_{k}=\frac{1}{k}\sum_{j=1}^{k}x_{j},\: k\in\mathbb{N}$, where $x=(x_{k})_{k=1}^{\infty}$ Show that $T$ is bounded if $1<p<\...
5
votes
4answers
277 views

Is there a null sequence that is in not in $\ell_p$ for any $p<\infty$?

Is $$\bigcup_{p<\infty}\ell_p=c_0 ?$$ At least one inclusion obvious: every $p$-summable sequence converges to zero.
3
votes
2answers
383 views

What is the norm of the operator $\phi: L^3[-2,2] \to \mathbb{C}$ defined by $ \phi(f)=\int_{0}^{1}e^xf(x-1)dx$?

I'm trying to figure out the norm $\|\phi\|$ of the functional $\phi: L^3[-2,2] \to \mathbb{C}$ defined by $$ \phi(f)=\int_{0}^{1}e^xf(x-1)\mathsf dx$$ but am struggling. I can't figure out how to ...