# Questions tagged [weak-lp-spaces]

This tag address to any question concerning weak-lp -spaces. which are larger spaces than classical lp-spaces. These spaces are particular cases of Lorentz-spaces.

40 questions
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### Show that $g_n(x)=ne^{-nx}$ is bounded, where $x\in\Omega=]0,1[$ for $n\in\Bbb{N}.$

Show that $g_n(x)=ne^{-nx}$ is bounded, where $x\in\Omega=]0,1[$ for $n\in\Bbb{N}.$ My trial I'm thought of it this way that if $g_n(x)=ne^{-nx}$ converges weakly in $L_1(\Omega)$ then it is bounded....
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### Counting Balls in $L^2_m[0,1]$

Setup and Thoughts to Date Let $B$ be the closed unit ball in ${L}^2_m[0,1]$, where $m$ is the Lebesgue measure, equipped with the weak topology. I know that $B$ is separable, so there exists a ...
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### Weak Star Convergence of $u_n=\sin(\frac{nx}{n+1})(1+\exp{(-n|y|)})$

Consider the sequence $$u_n=\sin(\frac{nx}{n+1})(1+\exp{(-n|y|)}) \text{ where } (x,y)\in I=[-1,1]\times[-1,1], n\in\mathbb{N}.$$ (a) Study the equicontinuity of $(u_n)$ ...
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### A proposition about bounded and weak-$L^2$ integrable fucntion

I have this proposition that I have no idea how to start. any hints helps. If $f\in L^{2,\infty}\cap L^{\infty}$ then $f\in L^p$ for $p\geq2$ and the second part asks me to generalize this result. ...
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### Embedding $L^p$ into weak $L^p$ and notion of boundary of $L^p$

I'm very much a novice to analysis, so this is a kind of conceptual question that is probably very much the wrong way to think about these spaces, but perhaps it fits my more topological brain. The ...
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### How to show that $C^\infty_0$ is not dense in $L^p_{weak} (\mathbb{R}^n)$?

Let $C^\infty_0$ denote the smooth functions on $\mathbb{R}^n$ which decay at infinity. Let $L^p_{\mathrm{weak}}(\mathbb{R}^n)$ denote the space of all functions $f:\mathbb{R}^n \rightarrow \mathbb{R}$...
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### Definition of weak $L^p$ spaces

Today I came across the notation $L_\text{weak}$ and I don't know what exactly it means. I can only find the weak Lp spaces , but here they don't use the same notation. Is this the same thing? I'm ...
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### Weak $L^{p}$ spaces are quasi-normed?
Let $(X,\mathcal{B}, \mu)$ be a measure space. Then for $0< p < \infty$ by definition $L^{p,\infty}(X,\mathcal{B}, \mu)$ is the class of all measureable functions $f$ such that \begin{...