# Tagged Questions

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 ...

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### Bound on $f_h(t) := \frac 1h \int_t^{t+h}f$

Given $f \in L^2(0,T;L^2(\Omega))$ define $$f_h(t) = \frac 1h \int_t^{t+h}f(s)\;\mathrm{d}s$$ for $t \in (0,T-h)$ and $f_h(t) = 0$ for $t > T-h$. In this paper (http://dml.cz/bitstream/handle/...
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### Why isn't $\ell^p$ locally convex for $0<p<1$?

I believe we have to distinguish the finite-dimensional from the infinite dimensional case. Regardless, if $0<p<1$, $\|x\|_p := (\sum |x_i|^p)^{\frac 1 p}$ is not a norm as it fails to satisfy ...
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### If $p$ is a distribution, what is the meaning of the claim $\nabla p\in L^p(\Omega)^d$

Let $d\in\mathbb N$ $\Omega\subseteq\mathbb R^d$ be open $\mathcal D(\Omega):=C_c^\infty(\Omega)$ $q\ge 1$ I've seen the following Lemma (without a proof) in a paper and don't understand how I ...
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### If $f \in L^p(\epsilon, T)$ for every $\epsilon > 0$, is $f \in L^p(0,T)$?

If $f \in L^p(\epsilon, T)$ for every $\epsilon > 0$, is it necessarily true that $f \in L^p(0,T)$? I don't see why not since the only point we have a problem may be at 0, but that is a null set. ...
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### Compute $\lim_{p\to 1+}\left\|f\right\|_{p}$ where $f\in L^{1}[0,1] \cap L^{2} [0,1]$

Let $f\in L^{1}[0,1] \cap L^{2} [0,1]$. Compute $\lim_{p\to 1+}\left\|f\right\|_{p}$. I think the result would be $\left\|f\right\|_{1}$,but I don't know how to prove it.
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### Let $f$ be a Lebesgue measurable function on $\Bbb{R}$ satisfying some properties, prove $f\equiv 0$ a.e.

Let $f$ be a Lebesgue measurable function on $\Bbb{R}$ satisfying: i) there is $p\in (1,\infty)$ such that $f\in L^p(I)$ for any bounded interval $I$. ii) there is some $\theta \in (0,1)$ ...
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### Linear operator on a dense subset of $L_p$ which is unbounded when extended to $L_p$

Let the linear operator $L:C^{\infty}_0([-1,1]) \rightarrow C(\{0\})$ be defined as $f(0)$ (evaluating a function in $C^{\infty}_0([-1,1])$ at $0.$ I would like to show that extending this to ...
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### Volterra Operator on Sobolev Space

I stumpled over the following result in a script: Let $1 \leq p < \infty$ and $f \in L_p[a,b]$. Define the Volterra operator as $$Vf(t) = \int_a^t f(s) ds.$$ Then we have $Vf \in W^{1, p}[a,b]$ ...
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### $\lim_{p \to \infty} \|f\|_p = \|f\|_{\infty}$: is convergence monotone when $\mu(X) \leq 1$?

This question is related to Exercise 3.3.7(b) in Cohn, Measure Theory, 2nd edition, which reads as follows: Let $(X, \mathcal A, \mu)$ be a finite measure space, and let $f$ be an $\mathcal A$-...
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### Is it natural how $L^p$ spaces measure local and global sizes the same?

This is a continuation of my question Spaces of functions similar to $L^p$ but with different local and global sizes. I have been bothered by the fact that the $L^p$ norm on $\mathbb R^n$, which is ...
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### Gradient of the solution for Poisson equation

Let $f(x)=(\nabla N *g)(x)$, where $N(x)=\frac{1}{|x|^{n-2}}$ for $n\geq 3$ is the Newtonian kernel, and $g\in L^1(\mathbb{R}^n)\cap L^{\infty}(\mathbb{R}^n)$. Then we can have that $f\in L^{\infty}$ ...
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### Isometric isomorphism between infinite sequences and $L^2[-1,1]$

Show that the set of all infinite sequences $(x_1, x_2, ...)$ where $$\sum\limits_{n=1}^\infty x_n ^2 < \infty$$ is isometrically isomorphic to $L^2[-1,1]$.
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### Convergence in $L^2(\Bbb R)$ implies convergence of the norms [closed]

If $||f_n-f||_{L^2(\mathbb{R})}\to 0$ is it always true that $||f||_{L^2(\mathbb{R})}=\lim_{n\to\infty}||f_n||_{L^2(\mathbb{R})}$?
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### Counter Example for Limit of $\|f\|_p$ in infinity convergence, When Measure space is not finite [closed]

I found a proof for this fact that limit of $\|f\|_p$ when $p \to \infty$ is $\|f\|_{\infty}$ in here when $f:X \to R$ and $X \in L^p$ measure space is finite. But I need a counter example for ...
Let's consider following equation: The problem $$\begin{cases} -\operatorname{div}\left( p\left(x\right) \nabla{u} \right) + q(x)u = f \quad\text{... on } \Omega \\ u = h(x) \quad\text{... on } \... 1answer 69 views ### Proof verification: Something similar to Riesz-Fischer Theorem Question: Suppose \{f_n\} converges to f in L^p(\mathbb{R}), 1\leq p<\infty. Prove that there is a subsequence \{f_{n_k}\} and g\in L^p(\mathbb{R}) so that f_{n_k}\to f a.e. and |f_{... 1answer 30 views ### Integration of periodic function f \in L^1([0, 2\pi]) While studying trigonometric series and L^p spaces I was wondering the following: Let's say we have a 2\pi-periodic function f \in L^1([0, 2\pi]) satisfying \int_{0}^{2\pi}f(x) \, dx = 0. Is ... 1answer 44 views ### Characterization of measures such that \frac{1}{x} \in L^1(H) Let H be a finite measure on (0,1). What conditions must H fulfill, such that \begin{equation*} \frac{1}{x} \in L^1(H),\ \ \ \frac{1}{1 - x} \in L^1(H) \end{equation*} I'm trying to characterize ... 1answer 48 views ### Convergence in L^p and convergence almost everywhere Why f_n converges to f in L^p space implies that exists subsequence of f_n converging to f almost everywhere? 2answers 59 views ### Constructing an L^2 space on the unit ring \mathcal{S^1} Revised Question: Starting with L^2[0,2\pi], does the canonical map$$[0,2\pi)\ni\theta\mapsto e^{i\theta}\in\mathcal{S^1}$$(with functions going across in the obvious way) turn L^2[\mathcal{S^1}]... 1answer 84 views ### Why are there no finitely additive measures on \ell_\infty for which the measure of every ball is positive and finite? As the question title suggests, why are there no finitely additive measures on \ell_\infty for which the measure of every ball is positive and finite? Here, we do not assume that the measure is ... 1answer 44 views ### Closed subset for L^2 strong and weak convergence I was trying to solve the following exercise. Let K a closed subset of \mathbb{R}.$$X:=\left\{f\in L^2[0,1]:f(x)\in K \mbox{ a.e. }\:x\in [0,1] \right\} Then: 1) $X$ is closed under strong ...
Consider a functional $G^+:L_p \to \mathbb{R}$. Here $L_p = L_p (X,\textbf{X}, \mu)$ is the collection of all integrable fns (f s.t. $\int \vert f \vert^p d \mu < \infty$ on the measure space \$(X,\...