Question

Proving a space is not Banach in $\ell^1$

score 545 · Answer 1 · 2026-04-14 13:45:09

545

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Proving a space is not Banach in $\ell^1$

Published on 14 Apr 2026 - 13:45

score 89 · Answer 2 · 2026-04-15 21:54:02

89

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$f_n \to f$ a.e. and $\lvert f_n(x) \rvert \le \lvert f(x) \rvert $ where $f_n, f \in L^\infty$

Published on 15 Apr 2026 - 21:54

score 366 · Answer 3 · 2026-04-15 16:22:04

366

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Application of Hölder to prove $L^q \subset L^p$ for $1\leq p\leq q<\infty$ with Lebesgue measure

Published on 15 Apr 2026 - 16:22

score 123 · Answer 4 · 2026-04-12 19:43:44

123

Views

limit and integral in the Lp space with Schwartz class.

Published on 12 Apr 2026 - 19:43

score 51 · Answer 5 · 2026-04-11 19:50:32

51

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Convergence in $\mathcal{S}'$ and $L^p$ spaces

Published on 11 Apr 2026 - 19:50

score 111 · Answer 6 · 2026-04-17 15:26:32

111

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Weak-* convergence in $L^\infty$ of a certain sequence of functions

Published on 17 Apr 2026 - 15:26

score 94 · Answer 7 · 2026-04-17 09:52:27

94

Views

Proving $\lim_{p\to 0} ||f||_p^p = \mu(\operatorname{supp}f)$

Published on 17 Apr 2026 - 9:52

score 77 · Answer 8 · 2026-04-15 10:20:02

77

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Prove that $G=\{f\in L_p(\mu): \int_\mathbb{R} f(t) dt = 0\}$ is a dense subset of $L_p(\mu)$, where $\mu$ is the lebesgue measure on $\mathbb{R}$.

Published on 15 Apr 2026 - 10:20

score 34 · Answer 9 · 2026-04-14 16:54:17

34

Views

Are these functions in $L^{1}(\mathbb{R}^{d})$?

Published on 14 Apr 2026 - 16:54

score 82 · Answer 10 · 2026-04-13 07:21:40

82

Views

Classic exercise of $L^p$ spaces but using uniform boundedness principle

Published on 13 Apr 2026 - 7:21

score 488 · Answer 11 · 2026-04-15 09:45:10

488

Views

Uniformly $L^{q}$ bounded and convergence in probability implies $L^{p}$ convergence

Published on 15 Apr 2026 - 9:45

score 43 · Answer 12 · 2026-04-14 01:01:12

43

Views

Showing a function $\notin L^1$

Published on 14 Apr 2026 - 1:01

score 107 · Answer 13 · 2026-04-13 00:08:36

107

Views

How to prove that the dual of $L^\infty$ is the set of bounded finitely additive measures?

Published on 13 Apr 2026 - 0:08

score 43 · Answer 14 · 2026-04-16 11:08:06

43

Views

Verify $||T||_{L(X,Y)}$ norm is given by $||T||_{L(X,Y)} = \sup_{x \in X, ||x||_X = 1} ||T(x)||_Y$.

Published on 16 Apr 2026 - 11:08

score 84 · Answer 15 · 2026-04-15 16:56:40

84

Views

$X_n$ is a sequence and uniformly distributed on $(-n,n)$. Does it converge in $L^p$?

Published on 15 Apr 2026 - 16:56

List Question

Proving a space is not Banach in $\ell^1$

$f_n \to f$ a.e. and $\lvert f_n(x) \rvert \le \lvert f(x) \rvert $ where $f_n, f \in L^\infty$

Application of Hölder to prove $L^q \subset L^p$ for $1\leq p\leq q<\infty$ with Lebesgue measure

limit and integral in the Lp space with Schwartz class.

Convergence in $\mathcal{S}'$ and $L^p$ spaces

Weak-* convergence in $L^\infty$ of a certain sequence of functions

Proving $\lim_{p\to 0} ||f||_p^p = \mu(\operatorname{supp}f)$

Prove that $G=\{f\in L_p(\mu): \int_\mathbb{R} f(t) dt = 0\}$ is a dense subset of $L_p(\mu)$, where $\mu$ is the lebesgue measure on $\mathbb{R}$.

Are these functions in $L^{1}(\mathbb{R}^{d})$?

Classic exercise of $L^p$ spaces but using uniform boundedness principle

Uniformly $L^{q}$ bounded and convergence in probability implies $L^{p}$ convergence

Showing a function $\notin L^1$

How to prove that the dual of $L^\infty$ is the set of bounded finitely additive measures?

Verify $||T||_{L(X,Y)}$ norm is given by $||T||_{L(X,Y)} = \sup_{x \in X, ||x||_X = 1} ||T(x)||_Y$.

$X_n$ is a sequence and uniformly distributed on $(-n,n)$. Does it converge in $L^p$?

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