Question

A function that is in the $L^p$ space for some $p \ge 1$ and uniformly continuous, has a limit of $0$ as $||x|| \rightarrow \infty.$

score 125 · Answer 1 · 2026-04-14 22:15:01

A function that is in the $L^p$ space for some $p \ge 1$ and uniformly continuous, has a limit of $0$ as $||x|| \rightarrow \infty.$

score 88 · Answer 2 · 2026-04-18 09:29:33

88

Views

$\int (f-g)\phi^{1/n}=0$ implies $f=g$ ae.

Published on 18 Apr 2026 - 9:29

score 118 · Answer 3 · 2026-04-14 09:14:20

118

Views

A property on $L^p$ and $L^q$ spaces

Published on 14 Apr 2026 - 9:14

score 66 · Answer 4 · 2026-04-12 18:27:27

66

Views

How to demonstrate that $\frac{\sqrt{n}}{1+\sqrt{nx}}$ converges in $L^{1}( (0,1),\mu)?$

Published on 12 Apr 2026 - 18:27

score 57 · Answer 5 · 2026-04-15 10:10:36

57

Views

Proof verification: To show that a function is not Lebesgue integrable.

Published on 15 Apr 2026 - 10:10

score 105 · Answer 6 · 2026-04-16 03:59:01

105

Views

Demonstrate that $\lim_{n \rightarrow \infty} \int_{\mathbb{R}} f(x)\sin(nx) ~ dx = 0 $ for $f \in L^{1}(\mathbb{R}).$

Published on 16 Apr 2026 - 3:59

score 133 · Answer 7 · 2026-04-12 16:03:52

133

Views

Approximation in $L^p$ spaces by continuous functions

Published on 12 Apr 2026 - 16:03

score 88 · Answer 8 · 2026-04-17 04:02:59

88

Views

Given a bounded sequence in $L^1([a,b])$, is $(t\mapsto \int_a^t f_n \,\mathrm d\lambda)_n$ equicontinuous?

Published on 17 Apr 2026 - 4:02

score 226 · Answer 9 · 2026-04-15 10:19:36

226

Views

Is the integral operator $I: L^1([0,1])\to L^1([0,1]), f\mapsto (x\mapsto \int_0^x f \,\mathrm d\lambda)$ compact?

Published on 15 Apr 2026 - 10:19

score 247 · Answer 10 · 2026-04-17 12:59:35

247

Views

$L^1_{\text{loc}}$, Frechet Space and norm-distance

Published on 17 Apr 2026 - 12:59

score 31 · Answer 11 · 2026-04-17 07:05:11

31

Views

To show that $\lim_{x\to\infty}\cfrac{1}{x^{1-1/p}}\int_0^x f(t)dt=0$ via Holder's inequality.

Published on 17 Apr 2026 - 7:05

score 49 · Answer 12 · 2026-04-17 11:40:46

49

Views

Let $f \in L^1(\mathbb{R})$ and $p$ a polynomial of deg $m$. Does $f / p \in L^1$ imply $x^m f / p \in L^1$ or vice versa?

Published on 17 Apr 2026 - 11:40

score 143 · Answer 13 · 2026-04-18 09:41:02

143

Views

To find a sequence on $L^1$-norm equal to 2, converging a.e. to a function of $L^1$norm equal to 1.

Published on 18 Apr 2026 - 9:41

score 31 · Answer 14 · 2026-04-16 22:21:57

31

Views

$\int_{E_n} |g|^q = \left|\int_E \chi_{E_n}\cdot \text{sgn}(g)\cdot g \cdot |g|^{q-1}\cdot |g|\right|$

Published on 16 Apr 2026 - 22:21

score 88 · Answer 15 · 2026-04-14 01:36:53

88

Views

For $p \ge 1$, demonstrate that $\log(x) \in L^{p}((0,1],\mu)$, where $\mu$ is the Lebesgue measure.

Published on 14 Apr 2026 - 1:36

List Question

A function that is in the $L^p$ space for some $p \ge 1$ and uniformly continuous, has a limit of $0$ as $||x|| \rightarrow \infty.$

$\int (f-g)\phi^{1/n}=0$ implies $f=g$ ae.

A property on $L^p$ and $L^q$ spaces

How to demonstrate that $\frac{\sqrt{n}}{1+\sqrt{nx}}$ converges in $L^{1}( (0,1),\mu)?$

Proof verification: To show that a function is not Lebesgue integrable.

Demonstrate that $\lim_{n \rightarrow \infty} \int_{\mathbb{R}} f(x)\sin(nx) ~ dx = 0 $ for $f \in L^{1}(\mathbb{R}).$

Approximation in $L^p$ spaces by continuous functions

Given a bounded sequence in $L^1([a,b])$, is $(t\mapsto \int_a^t f_n \,\mathrm d\lambda)_n$ equicontinuous?

Is the integral operator $I: L^1([0,1])\to L^1([0,1]), f\mapsto (x\mapsto \int_0^x f \,\mathrm d\lambda)$ compact?

$L^1_{\text{loc}}$, Frechet Space and norm-distance

To show that $\lim_{x\to\infty}\cfrac{1}{x^{1-1/p}}\int_0^x f(t)dt=0$ via Holder's inequality.

Let $f \in L^1(\mathbb{R})$ and $p$ a polynomial of deg $m$. Does $f / p \in L^1$ imply $x^m f / p \in L^1$ or vice versa?

To find a sequence on $L^1$-norm equal to 2, converging a.e. to a function of $L^1$norm equal to 1.

$\int_{E_n} |g|^q = \left|\int_E \chi_{E_n}\cdot \text{sgn}(g)\cdot g \cdot |g|^{q-1}\cdot |g|\right|$

For $p \ge 1$, demonstrate that $\log(x) \in L^{p}((0,1],\mu)$, where $\mu$ is the Lebesgue measure.

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