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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26 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 ...
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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+...
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 ...
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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 \...
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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)\...
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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 $...
14
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1answer
128 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 ...
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1answer
20 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: $1.$ ...
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1answer
23 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\...
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2answers
38 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 ...
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1answer
46 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 $...
2
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1answer
63 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.$...
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1answer
22 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/...
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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{...
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0answers
18 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
33 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)={...
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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 ...
3
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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
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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 ...
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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 \...
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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
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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 ...
1
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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 $u_n(x)=1/n*e^{-|x|/n}$...
2
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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 (...
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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 ...
2
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1answer
44 views

A question about distributions and Lp spaces [closed]

If all the partial derivatives of a distribution are $L^p$ functions for some $p$, is the distribution a regular distribution? Assume that an $L^1_{loc}$ function $f$ has all second partial ...
0
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1answer
25 views

Measure of Cauchy sequence

I consider $(f_n)_{n\in \mathbb N} \in L^p(\mu)$ and $\vert|f_n-f\vert|_p \rightarrow 0 $ for $ n \rightarrow \infty$ So I can conclude: Let $ \epsilon_k =2^{-k}$ $ \forall k \ \exists n_k: \mu(\{|...
0
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1answer
39 views

Boundedness in $H^1(\Omega) $ and tightness

Let $\Omega$ be a bounded subset of $\Bbb R^d$. And let $ (u_n)_n$ be a bounded sequence of the Sobolev space $H^1(\Omega)$. Question: Can we say that $ (u_n)_n$ is tight in $L^2(\Omega)$ namely: ...
2
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0answers
110 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
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1answer
57 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 ...
2
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1answer
23 views

$J$ is surjective but not injective on $L^{\infty}\to (L^{1})^{*}$

Let Let $X:=\{0, 1\}$ and $\mathcal{A}:=\mathcal{P}(X)$ and $\mu(\{0\})=1$ while $\mu(\{1\})=\infty$ I know that $f\in L^{1}(X,\mathcal{A}, \mu)$ iff $f\vert_{\{1\}}=0$ and $g \in L^{\infty}(X,\...
5
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2answers
41 views

A proper subspace of a normed vector space has empty interior clarification

So every proper subspace of a normed vector space has empty interior. I'm not asking for the proof, my problem is that this seems to me very strange. So if I have a normed vector space, in any proper ...
1
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1answer
29 views

Space $L^p$. I need to find a constant.

$L^p [0,1] = \{f: [0,1] \to \mathbb R$ ; $f$ measurable such that $\int_{0}^{1}|f|^p dx < + \infty\}$ and, $\parallel f \parallel_{p} = \int_{0}^{1} (|f|^{p})^{\frac{1}{p}}$. Let $F$ be a subspace ...
8
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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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1answer
44 views

Is $J$ injective or surjective where $Jf(g):=\int d\mu f g$

Let $X:=\{0, 1\}$ and $\mathcal{A}:=\mathcal{P}(X)$ and $\mu(\{0\})=1$ while $\mu(\{1\})=\infty$ I know that $f\in L^{1}(X,\mathcal{A}, \mu)$ iff $f\vert_{\{1\}}=0$ and $g \in L^{\infty}(X,\mathcal{A}...
0
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0answers
36 views

Relations between $L^{p}$ inner-products

The following full setup of this problem is a bit long, but the only problem is the final step of the proof. Setup: Let $\Omega$ be a domain in $\mathbb{R}$. Let $\phi_{1}, \phi_{2} \in H^{1}_{0, \...
1
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1answer
45 views

Convergence in $l^{p}$

Prove that, given $q \in [1,\infty] $, then $ l^{p} \hookrightarrow l^{q} $ for all $p \in [1, q]$. Consider the sequence $ x^{(n)}=\bigl( x^{(n)}_{k} \bigr)_{k \in \mathbb{N}_{0} } \ $ defined by \...
0
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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 ...
0
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2answers
39 views

Why $\lim_{m\rightarrow\infty}F_m(x)=\liminf_{m\rightarrow\infty}F_m(x)=f(x)$?

I am trying to understand a step in the proof of Use Fatou Lemma to show that $f$ takes real values almost everywhere. it is shown that $\lim_{m\rightarrow\infty}F_m(x)=\liminf_{m\rightarrow\infty}...
1
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0answers
37 views

Is my Interpretation of $L^{1}$ and $L^{\infty}$ correct in this case

Let $X:=[0,1]$ and $\mathcal{A}:=\{A \subseteq [0,1]: A \operatorname{or} A^{c} \operatorname{countable}\}$ and $\mu$ be the counting measure. I am asked to characterize both $L^{1}(X,\mathcal{A}, \...
1
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0answers
28 views

Find a functions $(f_{n})_{n}$ so that they converge in $L^{\infty}([-N,N])$ but not $L^{\infty}(\mathbb R)$

My idea: Let $f \equiv 1$ and set define $f_{n}\equiv 1_{[-n,n]}$ clearly then for any $N \in \mathbb N$ and $\epsilon>0$ there is $n_{0}$ large enough such that $\forall n \geq n_{0}$ it follows ...
1
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1answer
27 views

Showing $2(\|f\|_{L_p}^p+\|g\|_{L_p}^p)\le\|f-g\|_{L_p}^p+\|f+g\|_{L_p}^p$

I have the following question from a past qualifying exam: Given $2\le p<\infty$. Show that for any real-valued functions $f,g\in L_p(\mathbb{R})$, it holds that $$2\left(\left\|\frac{f}{2}\...
0
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0answers
16 views

weighted young inequality

Define weight $w(\theta) = (1+c \cos^2 \theta)^{s},$ $s>0$ $c$ some constant. Define $\|f\|_{L^p_w(\mathbb T)}^p=\int_{\mathbb T}|f(\theta)|^p w^{p}(\theta) d\theta $ Can we expect $\|f\ast ...
2
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3answers
241 views

How to prove this countable intersection is empty?

Let $(l_p,\|\cdot\|_p)$ be a normed space, for some fixed $p \in [1,\infty)$. Let $\{e_i\}_{i=1}^{\infty}$ be standard basis and $E_n=\{e_i\}_{i=n}^{\infty},\ n=1,2,3,...\ .$ Then how can I prove $\...
1
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1answer
49 views

Is it true that a sequence $ f_n \to f $ of measurable functions is bounded by a norm of $ L^p $ then $ f_n $ converges to $f$ in $ L^p $? [duplicate]

Is it true that a sequence $ f_n \to f $ of measurable functions is bounded by a norm of $ L^p $ then $ f_n $ converges to f in $ L^p $? Is this true? If so, prove, if not, a counter example. I just ...
0
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1answer
21 views

Splitting function from $L^{n/2}$

Let $V\in L^{n/2}$, $n\geq 3$. I want to show that for every $\varepsilon>0$, there are $||V_{1}||_{L^{n/2}}\leq \varepsilon$ and $V_{2}\in L^{\infty}$ sucht that $$V=V_{1}+V_{2}$$
0
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1answer
39 views

Let X be a measurable set with $ \mu (X) < \infty $ and $ 1 \leq p < \infty $… ($L^p$ spaces

Let X be a measurable set with $ \mu (X) < \infty $ and $ 1 \leq p < \infty $ Let $ (f_n)_{n \in \mathbb{N}} \subset X $ and $f \in L^{p}(X)$ with $\lim_{n \to \infty}||f_n - f|| = 0 $ Show ...
1
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1answer
31 views

Does translation operator preserve norm on weighted Lebesgue spaces?

We define weighted Lebesgue norm as follows: $\|f\|_{L_{w}^p}^p= \int_{\mathbb R^d} |f(x)|^p w(x) dx$ where $w$ is some nonnegative weight function (e.g., $ w(x)= (1+|x|^2)^{1/2}$ ) Fix $y\in \...
3
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1answer
37 views

Why are weights defined to be locally integrable?

My question is about why weights are defined to be locally integrable. I am trying to understand a statement in the paper "Poincare Inequalities and Neumann problems for the $p$-Laplacian" by David ...
3
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1answer
37 views

Why is boundedness sufficient for well-definedness

Let $p, q \in {]1,\infty[}$ where $\frac{1}{p}+\frac{1}{q}=1$ and define $J\colon L^{q} \to (L^{p})^{*}, f \mapsto \ell_{f}: L^{p} \ni g\mapsto \ell_{f}(g)=\int_{X}fg\,d\mu$ My question is related ...