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Triangle inequality fails in $L^{1,\infty}$

score 1.4k · Answer 1 · 2026-04-07 01:54:35

1.4k

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Triangle inequality fails in $L^{1,\infty}$

Published on 07 Apr 2026 - 1:54

score 489 · Answer 2 · 2026-04-13 06:38:03

489

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Real Analysis, Folland problem 6.1.11 $L^p$ spaces

Published on 13 Apr 2026 - 6:38

score 165 · Answer 3 · 2026-04-13 02:36:51

165

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Real Analysis, Folland Problem 6.1.16 $L^p$ spaces

Published on 13 Apr 2026 - 2:36

score 868 · Answer 4 · 2026-04-12 19:38:20

868

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Real Analysis, Folland Problem 6.2.17 Dual of $L^p$

Published on 12 Apr 2026 - 19:38

score 463 · Answer 5 · 2026-04-16 01:39:45

Let $\mu $ be a $\sigma $-finite measure. Show that for any $f\in L_p(\mu)$, $\|f\|_1=\sup\{ \int fg\, d\mu :\|g\|_\infty \leq 1\}$

score 64 · Answer 6 · 2026-04-15 04:11:18

64

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Where does this inequality come from: $\frac{|f(x) - f(x-h)|}{h} \le ||f'||_{L^\infty(x, x-1)}$?

Published on 15 Apr 2026 - 4:11

score 39 · Answer 7 · 2026-04-16 14:32:25

39

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Integrability of $f(x)\sqrt{\frac{1}{x}}$ for $f\in\mathcal L^2$ and $\|f\|_2=1$

Published on 16 Apr 2026 - 14:32

score 759 · Answer 8 · 2026-04-12 16:09:11

759

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Real Analysis , Folland Problem 6.1.5

Published on 12 Apr 2026 - 16:09

score 76 · Answer 9 · 2026-04-14 17:10:18

76

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Showing this function is $0$ a.e.

Published on 14 Apr 2026 - 17:10

score 80 · Answer 10 · 2026-04-14 05:50:01

80

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Covergence in $L^p$

Published on 14 Apr 2026 - 5:50

score 72 · Answer 11 · 2026-04-13 21:16:21

72

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Equality for $\lvert\lvert f\rvert\rvert_1$

Published on 13 Apr 2026 - 21:16

score 205 · Answer 12 · 2026-04-16 01:05:20

205

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$1\le p \lt \infty$ and $f_k$ nonnegative increasing. Then $f_k\to f$ in $L_p$ iff $\sup_k||f_k||_p \lt \infty$.

Published on 16 Apr 2026 - 1:05

score 503 · Answer 13 · 2026-04-11 09:16:14

503

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Does convergence almost everywhere to an $L^p$ function and existence of a weakly convergent subsequence guarantee weak convergence?

Published on 11 Apr 2026 - 9:16

score 471 · Answer 14 · 2026-04-13 10:47:11

471

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What is the motivation for $l^p$ space?

Published on 13 Apr 2026 - 10:47

score 2k · Answer 15 · 2026-03-29 07:22:12

2k

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Problem about a compact operator $T:l^p\rightarrow l^p$

Published on 29 Mar 2026 - 7:22

List Question

Triangle inequality fails in $L^{1,\infty}$

Real Analysis, Folland problem 6.1.11 $L^p$ spaces

Real Analysis, Folland Problem 6.1.16 $L^p$ spaces

Real Analysis, Folland Problem 6.2.17 Dual of $L^p$

Let $\mu $ be a $\sigma $-finite measure. Show that for any $f\in L_p(\mu)$, $\|f\|_1=\sup\{ \int fg\, d\mu :\|g\|_\infty \leq 1\}$

Where does this inequality come from: $\frac{|f(x) - f(x-h)|}{h} \le ||f'||_{L^\infty(x, x-1)}$?

Integrability of $f(x)\sqrt{\frac{1}{x}}$ for $f\in\mathcal L^2$ and $\|f\|_2=1$

Real Analysis , Folland Problem 6.1.5

Showing this function is $0$ a.e.

Covergence in $L^p$

Equality for $\lvert\lvert f\rvert\rvert_1$

$1\le p \lt \infty$ and $f_k$ nonnegative increasing. Then $f_k\to f$ in $L_p$ iff $\sup_k||f_k||_p \lt \infty$.

Does convergence almost everywhere to an $L^p$ function and existence of a weakly convergent subsequence guarantee weak convergence?

What is the motivation for $l^p$ space?

Problem about a compact operator $T:l^p\rightarrow l^p$

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