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.
5,894 questions
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$(X, \Sigma, \mu)$ finite measure space and $(f_n)_{n\in\mathbb{N}}$ is sequence in $L^p(X)$. If $f_n \to f$ uniformly on $X$, then $f \in L^p(X)$.
Help in understanding a proof written by a teacher on the following theorem.
Let $(X, \Sigma, \mu)$ be a finite measure space and let $(f_n)_{n\in\mathbb{N}}$ be a sequence of functions in $L^p(X)$.
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Step in Brezis' proof of chain rule for Sobolev spaces
I'm going through the proof of Corollary 8.11 in Brezis' Functional Analysis, Sobolev Spaces and Partial Differential Equations which states:
Let $G \in C^1(\mathbb{R})$ be such that $G(0) = 0$, and ...
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For $p\ge1$, if $f_n\to f$ in $L^p$ and $f_n$ has the pointwise limit, then do I have $f_n\to f$ almost everywhere?
For measure space $(S,\mathcal A,\mu)$ and $p\ge 1$, even if $\{f_n\}_{n\ge1}\subset L^p(S)$ converges to $f$ in $L^p(S)$, $\{f_n\}_n$ doesn't necessarily converge to $f$ a.e. (The reference is Does ...
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Is $C^\infty(U)\cap L^\infty(U)\cap W^{k,p}(U)$ dense in $W^{k,p}(U)\cap L^\infty(U)$?
I'm interested in whether smooth bounded functions are dense in Sobolev spaces. Specifically, letting $U\subset \Bbb R^n$ be open and bounded, is $C^\infty(U)\cap L^\infty(U)\cap W^{k,p}(U)$ dense in $...
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Reference request: Kadec-Klee property for $\ell^1$
I am looking for references (textbooks or articles) where it is shown that the Banach space $\ell^1$ has the Kadec–Klee property, i.e. that the weak topology and the norm topology coincide on the unit ...
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Elements belonging to $c_0$ but not to $l^p$ [closed]
Question. How can I relate a sequence in $c_0$ to a sequence in $l^p$? What i mean exactly can i write any sequence $(\mu_n)_n \in c_0$ as the products of two sequences one of them is in $l^p$ i.e.
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About Riesz representation theorem for $L^p$ spaces
First we did some reduction to only consider positive forms on $L^p(\Omega)$ with $\Omega$ a set of finite measure.
In the proof that we have been presented in class for this theorem, when we consider ...
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If $\nabla f \in L^p(D)$ for some $f \in L^1(D)$, does it follow that $f \in L^p(D)$?
Suppose that $D \subseteq \mathbb R^n$ is a bounded domain.
Suppose that $f \in L^1(D)$ is an integrable function such that $\nabla f \in L^p(D)$, for some $p > 1$.
Does it follow that $f \in L^p(D)...
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Given $g \in L^q$, the norm of the functionl $G(f) = \int fg \, d\mu$ on $L^p$ is $\lVert g\rVert_q$. [duplicate]
Let $(X, \mathscr{A}, \mu)$ be a measure space and let $g \in L^q$. Define a linear functional on $L^p$ given by $G(f) = \int fg \, d\mu$. I want to show that $||G|| = || g ||_q$.
My attempt: Hölder's ...
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On the completeness of $L_p$
There is this thing regarding this topic that confuses me, and I am afraid I am wrong about it, because as you know math has a lot of hidden details. Let $(X,\Sigma,\mu)$ be a measure space, $\mu(E)=0$...
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Are $(c_0^+, \prec)$ and $(c_0^+, \succ)$ isomorphic?
This is a segue from this question I posted the other day.
For $a, b:\mathbb N\to\mathbb R$, we write $a\prec b$ iff $\{n\in\mathbb N:a_n<b_n\}$ is cofinite. It is easy to see $\prec$ is a strict ...
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Uniform convergence of a functional.
Let $\mathcal{K} \subset L^1{([0,1];\mathbb{R})}$ be compact and define the following sequence $(f_n)_{n \in \mathbb{N}}$ via $f_n \colon \mathcal{K} \to \mathbb{R}$
$$
f_n(u)=\int_0^{1/n} |u(s)| \, \...
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If $\left\vert\int fg\right\vert\leq\left(\int|g|^p\right)^{\frac{1}{p}}$ for some $p\in(0,1)$, then $f=0$ a.e.
Let $f\in L_{\text{loc.}}^1(\Bbb{R}^d)$ such that for some $p\in (0,1)$,
$$\left\vert\int f(x)\,g(x)\,\mathrm dx\right\vert\leq\left(\int\vert g(x)\vert^p\,\mathrm dx\right)^{\frac{1}{p}}$$
for all $g\...
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Existence of a convergence sequence in weak-star topology
In a book concerning calculus of variations written by Giusti, I read the following
" Let $Q$ be a cube. For every $\lambda\in [0,1]$ there exists a sequence $\chi_h$ of characteristic functions ...
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Exercise 2.9 (b) - Topics in Banach Space Theory (Albiac, Kalton)
I'm trying to solve Exercise 2.9 from Albiac and Kalton's book Topics in Banach Space Theory. The exercise in question is the following:
Let $X$ be a Banach space.
(a) Show that for every $x^{**} \in ...
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