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 classes of measurable functions such that $|f|^p$ is $\mu$-integrable. Questions can be about properties of functions in these spaces, or when the ambient space in a problem is an $L^p$ space.

$L^p$ spaces are defined for $p\in(0,\infty]$ as follows. Let $(X,\mathcal F,\mu)$ be a measure space. For $p$ with $0 < p<\infty$ we write $L^p(X,\mu)$ ($L^p(X)$ or $L^p$ when there is no ambiguity), for the vector space of equivalence classes (for equality almost everywhere) of measurable functions $f$ such that $\int_X|f|^pd\mu$ is finite. When $1\leq p<\infty$ we endow $L^p$ with the norm $\lVert f\rVert_p:=\left(\int_X|f(x)|^pd\mu(x)\right)^{1/p}$, while for $0 < p < 1$ we write $\lVert f\rVert_p:=\int_X|f(x)|^pd\mu(x)$, which induces a metric on $L^p$.

For $p=+\infty$, $L^\infty$ is the space of equivalence classes of functions $f$ such that we can find a constant $C$ with $|f(x)|\leq C$ almost everywhere. Then $\lVert f\rVert_{\infty}$ is the infimum of constants $C$ satisfying the latter property, and is called the essential supremum of $|f|$.

These spaces are sometimes called Lebesgue spaces. If we work with counting measure on, for example, the set $\mathbb N$, we get sequence spaces $\ell^p$ and $\ell^\infty$ as special cases.

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Exercise regarding pointwise convergence and $L^p$ convergence

Let $f: \mathbb{R} \to \mathbb{R}$ continuos with compact support. ($supp f=K$) $ a_n \rightarrow a>0, b_n \rightarrow b , f_n(x) = f(a_nx+b_n)$ And let $g=f(ax+b)$, I want to show that $||f_n-g||_p \rightarrow 0$ for $p\in[1,\infty)$. It is…
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Is $l^2$ really a generalized Euclidean space?

Sorry for kind of clickbait title, but I am very interested in $l^2$ space - space of all square-summable (and ths convergent) sequences. First, since I have read that $l^2$ is (with a pinch of salt) something like generalized Euclidean space or…
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If $g\in L^2(X)$ satisfies the given condition, then $g\in L^{\infty}(X)$.

I got stuck with the following while going through Lemma $3.20$ from the book 'Ergodic Theory, Independence and Dichotomies' by Kerr and Li. Problem: Let $(X,\mu)$ be a probability measure space, and let $g\in L^2(X)$. If for all $h_1,h_2\in…
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Bounded sequence converging in $L^2$ to a bounded limit?

Assume that for all $k$, $\sup_{\omega \in \Omega}X_k(\omega) \leq 0$ with $\|X_k -X\|_ 2$ is converging in $L^2$. It seems obvious that $\sup(X) \leq 0$. It this true? What would I need to prove this, or is it a standard result? I wanted first to…
user2005142
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Does weak convergence in $L^2$ plus uniform convergence imply strong convergence in $L^2$?

If we have $\{u_n\}_{n \ge 1} \subseteq C^{1}[0,1]$ weakly converging to $u$ in $L^{2}([0,1])$, and $\{u_n\}_{n \ge 1}$ is uniformly convergent to some function $v$ on $[0,1]$, can we conclude that $\{u_n\}_{n \ge 1}$ strongly converges to $u$ in…
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Example of a function $f$ such that $f\cdot r\in L^2$ and $f'\cdot r\in L^2$ but $f'\cdot r^2\not\in L^2$

Let $\newcommand{\R}{\mathbb R}\R_+\overset{\text{Def.}}=[0,\infty[$. I am looking for an explicit example of a smooth function $f:\R_+\to\R$ such that all of the following hold: $f(r)\cdot r\in L^2(\R_+)$; $f'(r)\cdot r \in L^2(\R_+)$; $f'(r)\cdot…
Maximilian Janisch
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Convergence a.e. of series $\sum_{n=1}^\infty c_n\phi_n$ where $(\phi_n)_n$ is orthonormal and $(c_n)_n \in \ell^p$

I'm trying to understand a paragraph from the article Christ, Kiselev: Maximal functions associated to filtrations. After the proof of Theorem 1.1., the authors deduce a theorem of Menshov as a corollary: Let $(X, \mu)$ be a measure space and let…
mechanodroid
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All Hilbert spaces are isometric to $l^2(E)$ - how?

I was reading about Hilbert spaces and came across this line on Wikipedia: By choosing a Hilbert basis (i.e., a maximal orthonormal subset of $L^2$ or any Hilbert space), one sees that all Hilbert spaces are isometric to $ℓ^2(E)$, where $E$ is a…
Tereza Tizkova
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Is the space $W^{1,1}_0(0,1)$ an abstract $L^1$ space?

Let $W^{1,1}_0(0,1)$ be the space of functions on the interval $(0,1)$ that vanish at the boundary with the standard $W^{1,1}(0,1)$-norm. $$||f|| = \int_0^1 |f(x)| \, dx + \int_0^1 |f'(x)| \, dx.$$ Is this space a Banach lattice? It appears to…
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Derivative under integral sign (minimal hypothesis)

Let $f\in C^\infty(\mathbb R^{d_1}\times\mathbb R^{d_2})\,\cap\, W^{\infty,p}(\mathbb R^{d_1}\times\mathbb R^{d_2})$ for all $1
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Find all values of $p$, with $1\leq p \leq \infty $, for which $f\in L^p(X,\mu)$.

I am struggling on this qualifying exam question that I found. A hint is provided that says Fubini may be helpful here, but I can't see how to setup the problem to apply it. Here is the question: Let $(X,\mathcal{M},\mu) $ be a finite measure space.…
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The set $\{x=(x_n)\in\ell^2:\sum_n|\sin x_n|\lt\infty\}$ is dense in $\ell^2$

I want to show that the set $\{x=(x_n)\in\ell^2:\sum_n|\sin x_n|\lt\infty\}$ is dense in $\ell^2$. So we have to show for given $\epsilon\gt0$ and any $a\in\ell^2$, there is $x=(x_n)\in\ell^2$ such that $$||x-a||_2\lt\epsilon\text{ with…
Jimmy
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what is the meaning of $H_0^1(\mathbb{R})$ space?

So, I was looking at the definition for the $H_0^1(\Omega)$ space, and I was wondering that if $\Omega = \mathbb{R}$. This is really to embed the boundary conditions of my operator, $\mathcal{L}$ which is posed on the real line, but also has…
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Are solutions to this PDE bounded uniformly in $\mathbb{R}^n$

Let $u\in \dot{H}^1(\mathbb{R}^n)$ be a weak solution in dimension $n\geq 3$ to the following PDE, $$-\Delta u = \lambda \rho u$$ where $\rho\in L^{n/2}(\mathbb{R}^n)\cap L^{\infty}(\mathbb{R}^n)$ and $\lambda >0.$ Can we conclude that $u\in…
Student
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Behaviour of $L^2$ integrable smooth function at $\infty$

I have seen counterexamples of continuous functions $f$ such that they are in $L^2(\mathbb{R})$ and $|f(x)|\rightarrow \neq 0$ as $|x|\rightarrow \infty$. Now I was wondering what would happen if we have a smooth function . Will we have in this case…
whatever
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