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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0
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2answers
40 views

If $(f-\tilde f)g=0$ for all $g\in L^q$, then $f=\tilde f$

Let $(\Omega,\mathcal A,\mu)$ be a measure space, $p,q\ge1$ with $p^{-1}+q^{-1}=1$ and $f:\Omega\to\mathbb R$ be $\mathcal A$-measurable with $$\int|fg|\:{\rm d}\mu<\infty\;\;\;\text{for all }g\in ...
15
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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 ...
9
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2answers
427 views

What is the basis of the vector space $l^\infty$?

We know that every vector space has a Hamel basis and also every normed space need not have a Schauder basis. As the normed space $l^\infty$ is not Separable so can't have the Schauder basis, but on ...
0
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0answers
22 views

Convergence of a subsequence in $L^1([0, T])$

Let $T>0$. Let $(h_n)_{n\geq 1}\subseteq L^\infty([0, T])$ and $h\in L^\infty([0, T])$. Suppose that for all $f\in L^1([0, T])$ we have $$\lim_{n\rightarrow\infty}\int_0^Tf(t)h_n(t)dt = \int_0^Tf(t)...
16
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2answers
205 views
+100

Show that if the integral of function with compact support on straight line is zero, then $f$ is zero almost everywhere

I want to prove that that given $f:R^2 \rightarrow R$ which is continuous with compact support s.t the integral of $f$ for every straight line $l$ is zero ($\int f(l(t))\mathrm{d}t=0$) then $f$ is ...
-1
votes
2answers
39 views

Showing that $\vert \vert x \vert \vert _{p} \xrightarrow{ p \to \infty} \vert \vert x \vert\vert_{\infty}$ for $x \in \ell^{1}$

Show that $\vert \vert x \vert \vert _{p} \xrightarrow{ p \to \infty} \vert \vert x \vert\vert_{\infty}$ for $x \in \ell^{1}$ Let $0\neq x \in \ell^{1}$ and $p\in ]1,\infty[$ then $\vert \vert x \...
1
vote
1answer
31 views

Weak and classical derivatives: an overview

I am studying PDE's and we have defined the following notion of weak derivative: Given a domain $\Omega\subset\mathbb{R}^{n}$ a function $f\in L^1_{loc}(\Omega)$ is wealy differentiable with respect ...
0
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1answer
38 views

Why if $p\not = q$ we have $L^p(R^n) \not \subseteq L^q(R^n)$?

Why if $p\not = q$ we have $L^p(R^n) \not \subseteq L^q(R^n)$? This is a result present in my books, and I can't figure out really a nice proof about this. An example say that the function $u(x) = (1+...
0
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1answer
121 views

Is the space of all convergent sequences compact in the space of all bounded sequences($l^{\infty})$?

Is the space of all convergent sequences compact in the space of all bounded sequences($l^{\infty}$)? Argument: I think the answer is in yes because in particular if we consider a sequence of all ...
3
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2answers
37 views

Union of $l^p$, 0<p<1

Is it true that $$ \bigcup\limits_{0<p<1} l^p = l^1\quad?$$ The space $l^p$ is the space of the sequences $\{a_n\}_n$ with $\sum |a_n|^p <\infty.$ The one inclusion is obvious, as any ...
0
votes
2answers
26 views

How to show that $\vert \vert T \vert \vert = \sqrt{c}$ where $c:=\sum\limits_{j=1}^{\infty}\sum\limits_{k=1}^{\infty}\vert t_{jk}\vert^{2}$

Define $c:=\sum\limits_{j=1}^{\infty}\sum\limits_{k=1}^{\infty}\vert t_{jk}\vert^{2} <\infty$ and $T:\ell^{2} \to \ell^{2}$ where $(Tx)_{j}=\sum\limits_{k=1}^{\infty}t_{jk}x_{k}$ for all $j \in \...
19
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2answers
7k views

Hölder's inequality with three functions

Let $p,q,r \in (1,\infty)$ with $1/p+1/q+1/r=1$. Prove that for every functions $f \in L^p(\mathbb{R})$, $g \in L^q(\mathbb{R})$,and $h \in L^r(\mathbb{R})$ $$\int_{\mathbb{R}} |fgh|\leq \|f\|_p\...
0
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2answers
28 views

Showing well-definedness of an equivalence class in $L^{\infty}(\mathbb R)$

Let $\mathbb L:=\{ f\in L^{\infty}(\mathbb R): \operatorname{there exists a representative}$ $\widetilde{f}$ so that $\lim\limits_{x \to \infty}f(x):=\lim\limits_{x \to \infty}\widetilde{f}(x)\...
0
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2answers
27 views

For what $p \in [1, + \infty]$ and $a \in R$ the function $u(x)=(1+|x|)^{-a}$ defined on $R^n$ verify $||u||_{L_p}< \infty$?

Has been 7 years from my last $L_p$ spaces experiences, now I have an exam about this. I have difficulties with the very first exercise: For what $p \in [1, + \infty]$ and $a \in R$ the function $...
0
votes
1answer
33 views

Notions of strong and weak convergence for $L^{2}(\mathbb R, \lambda)$ where $f_{n}(x)=1_{[n,n+1]}(x)$

Let $(f_{n})_{n} \subseteq L^{2}(\mathbb R, \lambda)$ where $f_{n}(x)=1_{[n,n+1]}(x)$. Investigate: $1. \lambda-$a.e. convergence $2. $ weak convergence $3.$ strong convergence My ideas: $1.$ ...
2
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1answer
65 views

Condition to have an almost everywhere property

Let $T>0$ and $g\in L^1([0,T])$ a non-negative function. Suppose that for all continuous and non-negative functions $f:[0,T]\rightarrow \mathbb R$ we have $$\int_0^Tf(t)g(t)dt \leq \int_0^Tf(t)dt.$...
1
vote
2answers
39 views

Example of $(L^1)^* \neq L^\infty$ from Exercise 6.12 in Rudin's RCA

This is Exercise 6.12 from Rudin's RCA. Let $\mathscr{M}$ be the collection of all sets $E$ in the unit interval $[0,1]$ such that either $E$ or its complement is at most countable. Let $\mu$ be the ...
0
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1answer
47 views

Extend inequality for $L^2$-inner product

Let $(E,\mathcal E,\mu)$ be a probability space and $f\in L^2(\mu)$. Assume there is a $c\ge0$ with that we know that $$|\langle f,g\rangle_{L^2(\mu)}|\le c\left\|g\right\|_{L^2(\mu)}\tag1$$ for all $...
1
vote
1answer
24 views

Almost everywhere convergence of functions which verify a property on the set $C_b$

Let $T>0$. Let $(h_n)_{n\geq 1}\subseteq L^\infty([0, T])$ and $h\in L^\infty([0, T])$. Suppose that for all continuous and bounded functions $f:[0, T]\rightarrow \mathbb R$ we have $$\lim_{n\...
1
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1answer
24 views

Struggle with dense set notation

In class we had the Proposition about density of compactly supported continuous functions $C_c(X)$ in $L^p(X)$ (If you do not know the Prop. see e.g.: https://planetmath.org/...
1
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2answers
63 views

Proving continuity of a function in a normed space

I have the normed space $ \ell^1 =\{(x_n) : \sum_{n=1}^\infty |x_n| < \infty\}$ with the norm defined by $\|x\|_1 = \sum_{n=1}^\infty|x_n| $, and a function $$f(x):=\sum_{n=1}^\infty x_n\sin(n) .$$ ...
8
votes
1answer
179 views

Does multiplication by a test function stay in a Sobolev space?

Let $u \in D^{1,\vec{p}}(\Omega)$ and $\phi \in C_c^{\infty}(\Omega)$. Then do we necessarily have $u\phi \in D^{1,\vec{p}}(\Omega)$? My attempt What we need to show is that $\partial_i (u \phi) \...
0
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2answers
69 views

My assumtion that $\lim\limits_{\vert x \vert \to \infty} f(x)=0$ if $f \in L^{p}$ so why does the following hold

I recently found out that $f \in L^{p}(\mathbb R)$ does not necessarily imply that $\lim\limits_{\vert x \vert \to \infty} f(x)=0$. Take for example, $f(x)=\begin{cases} n, \text{if } x \in [n,n+\frac{...
2
votes
1answer
48 views

Suffice condition for Weak convergence in $L^2$

Define $$ \langle f,g\rangle=\int_{\mathbb{R}}f(x)g(x)dx.$$ I know that in order to show that a sequence $\{f_n\}\in L^2(\mathbb{R})$ converge weakly to $f\in L^2(\mathbb{R})$, its suffice to show (...
3
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3answers
73 views

Hint: Showing $f_{n}(x):=\frac{n\sqrt{x} \sin{(x)}}{1+nx^{2}}$ converges in $L^{p}$ to $x^{-\frac{3}{2}}\sin{(x)}$

Let $f_{n}:]0, \infty[\to \mathbb R$ and $f_{n}(x):=\frac{n\sqrt{x} \sin{(x)}}{1+nx^{2}}$ Show that $f_{n}$ converges in $L^{p}$ to $f$ where $f(x):=x^{-\frac{3}{2}}\sin{(x)}$ and $p \in [1,2[$ My ...
1
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0answers
19 views

Is minimizer in reproducing kernel hilbert space the same as in L2 space?

In detail, $\mathcal{H}$ is a reproducing kernel hilbert space (RKHS), and $f^*$ is fixed. Problem 1 is $$f(x) \in \underset{g\in \mathcal{H}}{arg min} \Vert f^*(x) -g(x)\Vert_{\mathcal{H}}$$ ...
2
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0answers
34 views

Quadratic form in $L^{2}$ is closed

Let $V\in L_{\text{loc}}^{1}(\mathbb{R}^{n})$ and consider the quadratic form $q(u):=\langle Vu, u\rangle_{L^{2}(dx)}$. I want to show that this form is closed in $L^{2}$ with $\operatorname{dom}(q)={...
0
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0answers
53 views

On separability of $L^2$ and others.

I have two related questions: Is $L^2(X,\mu)$ separable if $\mu$ is a probability measure that is Lebesgue except it may have finitely many atoms? If I want to prove that a function is continuous ...
1
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0answers
39 views

Uniform bound for a family of linear operators on $L^2$

Let $\phi,\psi$ be functions on $\mathbb{R}^d$ with $\int\phi=\int\psi=0$ and $$|\phi(x)|,|\psi(x)|\lesssim(1+|x|)^{-(d+1)} \quad\text{and}\quad |\nabla\phi(x)|,|\nabla\psi(x)|\lesssim(1+|x|)^{...
1
vote
1answer
60 views

Sequence spaces $l^p$

I am trying to solve a question as regards $l^p$. Show that $$\lim_{j\rightarrow\infty}\sum_{n=1}^{\infty}\frac{x_n}{j+n}=0$$ for all $(x_1,x_2,...)\in l^2.$ I came up with some idea (which is not ...
3
votes
3answers
59 views

Proof Correct? Show that $\sum_{n \in \mathbb N}s_{n}t_{n}<\infty \Rightarrow s \in \ell^{1}$

Let $(s_{n})_{n}\subseteq \mathbb R$ and $c_{0}$ the space of null sequences. Show that $\sum_{n \in \mathbb N}s_{n}t_{n}<\infty \operatorname{for all }t\in c_{0} \Rightarrow s \in \ell^{1}$. My ...
3
votes
0answers
50 views

If $f,h,g\in L^{2}(\mathbb R^2)$, then $\int_{\mathbb R^3}\vert f(x,y)g(y,z)h(z,x)\vert d(x,y,z)\leq\|f\|_2\|g\|_2\|h\|_2$ [duplicate]

Let $f,h,g\in L^{2}(\mathbb R^2).$ Prove the inequality $$\int_{\mathbb R^3}\vert f(x,y)g(y,z)h(z,x)\vert d(x,y,z)\leq\|f\|_2\|g\|_2\|h\|_2.$$ Using Fubini's Theorem and the Cauchy-Schwarz ...
18
votes
4answers
12k views

Why is $l^\infty$ not separable?

My functional analysis textbook says "The metric space $l^\infty$ is not separable." The metric defined between two sequences $\{a_1,a_2,a_3\dots\}$ and $\{b_1,b_2,b_3,\dots\}$ is $\sup\limits_{i\...
1
vote
0answers
22 views

Embedding for $C_{0}(\Omega)$ to $L^{p}(\Omega)$

Let $\Omega\subset\mathbb{R}^{N}$ be a bounded domain and $f_{1}:\Omega\to\mathbb{R}$ be an element of $C_{0}(\Omega)$. Define $C_{0}(\Omega) := \{f\in C(\overline{\Omega})\,|\, f|_{\partial\Omega}=0 \...
4
votes
2answers
4k views

Prove that sequence space $\ell_p(\mathbb R)$ is separable

Problem: Prove that metric space $\left \langle \ell_p(\mathbb R), d_p(x,y)=(\sum_{i=1}^{\infty} |x_i|^p)^\frac{1}{p} \right \rangle$ is separable. Where $\ell_p(\mathbb R)=\left \{ (x_1,x_2,...,...
2
votes
2answers
330 views

How to conclude that $\ell_\infty$ is not separable from this exercise?

I have done an exercise that goes like this: Consider the operator $\Phi: \ell_1\to(\ell_\infty)'$ that associates each $x=(x_j)_j\in\ell_1$ to $\Phi (x)\in (\ell_\infty)'$ given by $\Phi(x)(y)=\...
2
votes
0answers
258 views

Dual space of $l^p$ is $l^q$.

I was reading functional analysis from Kreyszig. While proving that dual space of $l^p$ is $l^q$, I came across a doubt. I have attached the screenshot. In this they are applying f to $x_n$ but for ...
6
votes
2answers
3k views

Prove the dual space of $l^p$ is isomorphic to $l^q$ if $\frac{1}{q}+\frac{1}{p}=1$

Prove the dual space of $\ell^p$ is isomorphic to $\ell^q$ if $\frac{1}{q}+\frac{1}{p}=1$ ($1<p<\infty$) Define a map $J:\ell^q \to (\ell^p)'$ such that $Jy(x)=\sum_{k=1}^\infty x_ky_k,x\in \...
2
votes
2answers
303 views

$(e_1,e_2,..)$ is not a Schauder basis of $\ell^\infty$ [duplicate]

Show that $(e_1,e_2,...)$ is not a Schauder basis of $\ell^\infty$ where $e_i$ is the vector in $\mathbb R^\infty$ with 1 in the ith coordinate and 0 elsewhere and $\ell^\infty=\{(x_1,x_2,...)|x_i\in \...
2
votes
2answers
624 views

schauder basis for $\ell_\infty$ [duplicate]

I know that $\ell_\infty$ is not separable, therefore has no Schauder basis. However I cannot understand why the set $\{e_1, e_2, e_3, \dotsc \}$ where $e_1=(1,0,0,\dotsc), e_2=(0,1,0,0,\dotsc), \...
5
votes
2answers
290 views

Why is $(e_n)$ not a basis for $\ell_\infty$?

Let $(e_n)$ (where $ e_n $ has a 1 in the $n$-th place and zeros otherwise) be unit standard vectors of $\ell_\infty$. Why is $(e_n)$ not a basis for $\ell_\infty$? Thanks.
15
votes
0answers
146 views

$\frac{1}{p}+\frac{1}{q}=1$ vs $\sum_{n=0}^\infty \frac{1}{p^n}=q$

It just occurred to me that conjugate exponents, i.e. $p,q\in(1,+\infty)$ such that $$\frac{1}{p}+\frac{1}{q} =1$$ also satisfy the relations: $\sum_{n=0}^\infty \frac{1}{p^n}=q;$ $\sum_{n=0}^\infty \...
1
vote
3answers
127 views

Definition of the periodic $L^p$ space on torus

In his Real Analysis, Folland uses the notation $L^p({\mathbb T}^n)$ (where $\mathbb{T}^n$ denotes the n-dimensional torus) is used before Hausdorff measure is introduced. (See for instance Chapter 8: ...
1
vote
2answers
49 views

$\mathscr{S}(\mathbb{R}) \text {is dense in } L^{2}(\mathbb{R})$

This lemma was given by my professor in my notes of mathematical methods in physics. I believe with this lemma I can say that any function in the Schwartz space is $L^2$ integrable. However my ...
2
votes
0answers
111 views

Transformation of an $L^1$ function by a homomorphism of measure space is also an $L^1$ function.

Let $(X,\mathscr{B},\mu)$ and $(Y,\mathscr{D},\nu)$ be two complete measure spaces and $\alpha:\mathscr{D}\rightarrow \mathscr{B}$ be a homomorphism, i.e., a map satisfying $\alpha(A_1\cup A_2)=\...
1
vote
2answers
34 views

On convergence in $L^p$

I was trying to understand an exercise about convergence in $L^p$. I was asked to investigate the punctual convergence and the $L^p(\Bbb R)$ convergence, $1\le p\le +\infty$, of $u_n(x)=1/n*e^{-|x|/n}$...
5
votes
3answers
212 views

Use Fatou Lemma to show that $f$ takes real values almost everywhere.

Let $(f_n)$ be a sequence in $L^p$ such that for each positive integer $n$, $ \| f_{n+1}-f_n\|_{p} <\frac 1{2^n} $. Define $f: X \to [0,\infty]$ with $$ f(x)= \sum_{n=1}^\infty| f_{n+1}(x)-...
0
votes
2answers
50 views

Understanding a step in a proof of $L^p(X)\subseteq L^r(X)\subseteq L^1(X)$

I am trying to understand this proof to the question: $L^p$ and $L^q$ space inclusion. Here is the linked answer I am reading: There is a easy way to show that. Suppose that $p<q$ and X a space ...
1
vote
1answer
28 views

Is $L^{N}(\Omega) $ closed in $L^{2}(\Omega)$?

If $\Omega$ is a space of finite measure, then it is well known that $L^{N}(\Omega) \subseteq L^{2}(\Omega)$ for $N \in [2, \infty]$. I want to know if the image of this embedding is closed in $\left(...
1
vote
3answers
69 views

How to prove the “bang-bang” characterization on the alignment between $L_1[0,1]$ and $L_\infty[0,1]$?

I am following Luenberger's book "Optimization by Vector Space Methods". The author's solution to Example 2 on Page 124 obviously has utilized the following result on the characteristic of alignment ...