List questions from lp-spaces

Published on 02 Apr 2026 - 9:14

56

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Show that $L^1(\Bbb R)$ is not reflexive

Published on 20 Nov 2019 - 11:57

16

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Show that $\mathbb P ' \approx \mathbb P$ exists with bounded density so that $Y^{1},...,Y^{n}\in L^{1}(\mathbb P^{'})$

Published on 30 Mar 2026 - 11:37

92

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$T_{a}(\text{ball}(\ell^p))$ is compact, where $T_a\colon\ell^p\to\ell^p$, $(T_a x)_{n}:=a_{n}x_{n}$ and $\mathbb{C}\ni a_n\to0$ as $n\to\infty$

Published on 26 Mar 2026 - 19:02

191

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A funny exercise with bounded Lipschitz domain and Sobolev spaces

Published on 02 Apr 2026 - 2:00

24

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if $\lim_{n\to\pm\infty} |n|^k\widehat{f}(n)=0$ then $f \in C^\infty$ (in $L^1(\mathbb T)$

Published on 01 Apr 2026 - 14:24

90

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Give a counter-example to show translation is not continuous on $L^p$ when $p = \infty$

Published on 21 Nov 2019 - 8:41

84

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Proving that the set of continuous real functions is dense in the norm $L_{1}$ in the set of integrable functions

Published on 26 Mar 2026 - 0:53

269

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Rudin's exercise: showing that $(L^\infty)^*\neq L_1$

Published on 25 Mar 2026 - 14:37

63

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Prove that the following set has full measure (i.e. has probability one):

Published on 28 Mar 2026 - 10:15

67

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Does $\|f\|_{L^2}\,\le\,C(\|f'\|_{L^2} + \|xf\|_{L^2})$ for $f\in H^1(-1,1)$?

Published on 02 Apr 2026 - 3:38

38

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Finding $p>0$ for which $\log x\in L^p$

Published on 03 Apr 2026 - 15:35

45

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Confusion about $L^1$ and $\mathcal{L}^1$

Published on 23 Feb 2026 - 7:53

63

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Under what conditions do $L^p(\mu) = L^q(\mu)$ for $0<p<q$

Published on 23 Nov 2019 - 20:45

39

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Relationship between $l^{2}(\mathbb{N})$ and $l^{2}(\mathbb{R})$

Published on 25 Nov 2019 - 4:46

List Question

Inclusion of Lorentz Space

Show that $L^1(\Bbb R)$ is not reflexive

Show that $\mathbb P ' \approx \mathbb P$ exists with bounded density so that $Y^{1},...,Y^{n}\in L^{1}(\mathbb P^{'})$

$T_{a}(\text{ball}(\ell^p))$ is compact, where $T_a\colon\ell^p\to\ell^p$, $(T_a x)_{n}:=a_{n}x_{n}$ and $\mathbb{C}\ni a_n\to0$ as $n\to\infty$

A funny exercise with bounded Lipschitz domain and Sobolev spaces

if $\lim_{n\to\pm\infty} |n|^k\widehat{f}(n)=0$ then $f \in C^\infty$ (in $L^1(\mathbb T)$

Give a counter-example to show translation is not continuous on $L^p$ when $p = \infty$

Proving that the set of continuous real functions is dense in the norm $L_{1}$ in the set of integrable functions

Rudin's exercise: showing that $(L^\infty)^*\neq L_1$

Prove that the following set has full measure (i.e. has probability one):

Does $\|f\|_{L^2}\,\le\,C(\|f'\|_{L^2} + \|xf\|_{L^2})$ for $f\in H^1(-1,1)$?

Finding $p>0$ for which $\log x\in L^p$

Confusion about $L^1$ and $\mathcal{L}^1$

Under what conditions do $L^p(\mu) = L^q(\mu)$ for $0<p<q$

Relationship between $l^{2}(\mathbb{N})$ and $l^{2}(\mathbb{R})$

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