Question

Show that $M:=\{e^{-\alpha \mid x \mid}\varphi : \varphi \in C_c^\infty (\Bbb R^n)\}$ is dense in $\mathrm L ^2 (\Bbb R^n, \lambda ^n)$

score 1.2k · Answer 1 · 2017-01-15 12:20:50

1.2k

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Show that $M:=\{e^{-\alpha \mid x \mid}\varphi : \varphi \in C_c^\infty (\Bbb R^n)\}$ is dense in $\mathrm L ^2 (\Bbb R^n, \lambda ^n)$

Published on 15 Jan 2017 - 12:20

score 49 · Answer 2 · 2017-01-15 18:26:39

49

Views

$Z_n\stackrel{L^2}{\to} Z$ imply $Z_n^2\stackrel{L^1}{\to} Z^2$

Published on 15 Jan 2017 - 18:26

score 408 · Answer 3 · 2026-03-31 20:25:44

408

Views

Convergence of a sequence in $L^2$, mollifiers

Published on 31 Mar 2026 - 20:25

score 68 · Answer 4 · 2017-01-16 15:41:44

68

Views

$C(\mathbb{R})\cap L^p(\mathbb{R})$ is dense in $L^p(\mathbb{R})$

Published on 16 Jan 2017 - 15:41

score 446 · Answer 5 · 2026-04-04 18:54:03

446

Views

Uniqueness of weak limit in $H_0^1({\Omega})$ and strong limit in $L^2(\Omega)$

Published on 04 Apr 2026 - 18:54

score 95 · Answer 6 · 2017-01-18 04:44:20

95

Views

uniform continuity and invertiblity of a map on $\ell^p$ -spaces?

Published on 18 Jan 2017 - 4:44

score 309 · Answer 7 · 2026-03-28 14:05:42

309

Views

Direct sum of Hilbert Space $\ell^{2}$

Published on 28 Mar 2026 - 14:05

score 68 · Answer 8 · 2026-04-04 18:54:09

68

Views

Doubt in Moorey's ineqality given in Sobolev space .

Published on 04 Apr 2026 - 18:54

score 61 · Answer 9 · 2017-01-18 21:51:54

61

Views

$L^p$ convergence follows from $L^2$ convergence plus boundedness of $L^\infty$?

Published on 18 Jan 2017 - 21:51

score 456 · Answer 10 · 2026-03-29 15:54:58

456

Views

How to prove that the derivative of heaviside function is not an element of $L^2$

Published on 29 Mar 2026 - 15:54

score 359 · Answer 11 · 2026-03-31 15:16:55

359

Views

Injectivity of a multiplication operator implies that the corresponding function is supported almost everywhere

Published on 31 Mar 2026 - 15:16

score 935 · Answer 12 · 2017-01-19 19:30:23

935

Views

Convergence in $L^p$ and in $L^{\infty}$

Published on 19 Jan 2017 - 19:30

score 126 · Answer 13 · 2017-01-19 20:49:06

126

Views

Ultraproduct of partitions of an $L^p$-space isomorphic to itself?

Published on 19 Jan 2017 - 20:49

score 168 · Answer 14 · 2017-01-20 02:53:08

168

Views

Show that $(a_n)_n\in l^\infty$ if $(a_nx_n)\in l^p$ for each $(x_n)\in l^p$

Published on 20 Jan 2017 - 2:53

score 476 · Answer 15 · 2017-01-21 16:25:41

476

Views

Precise statement of the Lebesgue differentiation theorem

Published on 21 Jan 2017 - 16:25

List Question

Show that $M:=\{e^{-\alpha \mid x \mid}\varphi : \varphi \in C_c^\infty (\Bbb R^n)\}$ is dense in $\mathrm L ^2 (\Bbb R^n, \lambda ^n)$

$Z_n\stackrel{L^2}{\to} Z$ imply $Z_n^2\stackrel{L^1}{\to} Z^2$

Convergence of a sequence in $L^2$, mollifiers

$C(\mathbb{R})\cap L^p(\mathbb{R})$ is dense in $L^p(\mathbb{R})$

Uniqueness of weak limit in $H_0^1({\Omega})$ and strong limit in $L^2(\Omega)$

uniform continuity and invertiblity of a map on $\ell^p$ -spaces?

Direct sum of Hilbert Space $\ell^{2}$

Doubt in Moorey's ineqality given in Sobolev space .

$L^p$ convergence follows from $L^2$ convergence plus boundedness of $L^\infty$?

How to prove that the derivative of heaviside function is not an element of $L^2$

Injectivity of a multiplication operator implies that the corresponding function is supported almost everywhere

Convergence in $L^p$ and in $L^{\infty}$

Ultraproduct of partitions of an $L^p$-space isomorphic to itself?

Show that $(a_n)_n\in l^\infty$ if $(a_nx_n)\in l^p$ for each $(x_n)\in l^p$

Precise statement of the Lebesgue differentiation theorem

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