Question

For $1 \leq r < p < \infty$ prove the continuous injection of $L^p([0, 1])$ into $L^r([0, 1])$.

score 112 · Answer 1 · 2014-07-20 15:22:21

112

Views

For $1 \leq r < p < \infty$ prove the continuous injection of $L^p([0, 1])$ into $L^r([0, 1])$.

Published on 20 Jul 2014 - 15:22

score 249 · Answer 2 · 2026-03-25 06:00:30

249

Views

Prove this function is absolute continuous and Lipschitz of order $\alpha$

Published on 25 Mar 2026 - 6:00

score 76 · Answer 3 · 2014-07-21 16:43:30

76

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Are these two expression square integrable?

Published on 21 Jul 2014 - 16:43

score 67 · Answer 4 · 2014-07-21 20:06:09

67

Views

Show that $\int_E f (x, y) dx$ is differentiable with respect to $y$ and $\frac{d}{dy}\int_E f(x,y)dx=\int_E \frac{d}{dy}f(x,y)dx.$

Published on 21 Jul 2014 - 20:06

score 335 · Answer 5 · 2014-07-22 00:12:14

335

Views

Let $(X,μ)$ be a measure space. Find a necessary and sufficient condition on $(X,μ)$ that $L_q(E) ⊂ L_p(E)$ for all $1 ≤ p < q ≤ ∞.$

Published on 22 Jul 2014 - 0:12

score 181 · Answer 6 · 2014-07-22 13:31:32

181

Views

Integration by substitution and separation of variables.

Published on 22 Jul 2014 - 13:31

score 61 · Answer 7 · 2014-07-22 14:34:58

61

Views

Prove that $f\ast g$ is defined a.e., integrable, and such that $∥f\ast g∥_1 ≤ ∥f∥_1 · ∥g∥_1$

Published on 22 Jul 2014 - 14:34

score 184 · Answer 8 · 2014-07-22 18:46:24

184

Views

Find $\lim_{n \rightarrow \infty}$$\int_0^n(1 + \frac{-x}{n})^n\cos(\frac{x}{\sqrt{n}})e^{x/2}dx$

Published on 22 Jul 2014 - 18:46

score 516 · Answer 9 · 2014-07-23 13:22:32

516

Views

Inequality involving Jensen (Rudin's exercise)

Published on 23 Jul 2014 - 13:22

score 114 · Answer 10 · 2014-07-23 22:21:06

114

Views

Not lebesgue integrable function?

Published on 23 Jul 2014 - 22:21

score 55 · Answer 11 · 2014-07-24 12:06:24

55

Views

How do the inner products on $L^2$ look like?

Published on 24 Jul 2014 - 12:06

score 123 · Answer 12 · 2026-03-31 15:52:34

123

Views

Finite Measure Space: Integral Closure = Bochner Integral

Published on 31 Mar 2026 - 15:52

score 565 · Answer 13 · 2026-03-27 23:31:56

565

Views

About the $\lim_{n \to+\infty} \frac{1}{n}\int_0^1 \log(1+e^{nf(x)})\,dx$ (Rudin's exercise)

Published on 27 Mar 2026 - 23:31

score 536 · Answer 14 · 2014-07-26 03:50:10

536

Views

Lebesgue integral of $\chi_{\mathbb{Q}}: \mathbb{R} \rightarrow \mathbb{R}$

Published on 26 Jul 2014 - 3:50

score 50 · Answer 15 · 2014-07-26 18:10:19

50

Views

Problem involving decomposition of measures

Published on 26 Jul 2014 - 18:10

List Question

For $1 \leq r < p < \infty$ prove the continuous injection of $L^p([0, 1])$ into $L^r([0, 1])$.

Prove this function is absolute continuous and Lipschitz of order $\alpha$

Are these two expression square integrable?

Show that $\int_E f (x, y) dx$ is differentiable with respect to $y$ and $\frac{d}{dy}\int_E f(x,y)dx=\int_E \frac{d}{dy}f(x,y)dx.$

Let $(X,μ)$ be a measure space. Find a necessary and sufficient condition on $(X,μ)$ that $L_q(E) ⊂ L_p(E)$ for all $1 ≤ p < q ≤ ∞.$

Integration by substitution and separation of variables.

Prove that $f\ast g$ is defined a.e., integrable, and such that $∥f\ast g∥_1 ≤ ∥f∥_1 · ∥g∥_1$

Find $\lim_{n \rightarrow \infty}$$\int_0^n(1 + \frac{-x}{n})^n\cos(\frac{x}{\sqrt{n}})e^{x/2}dx$

Inequality involving Jensen (Rudin's exercise)

Not lebesgue integrable function?

How do the inner products on $L^2$ look like?

Finite Measure Space: Integral Closure = Bochner Integral

About the $\lim_{n \to+\infty} \frac{1}{n}\int_0^1 \log(1+e^{nf(x)})\,dx$ (Rudin's exercise)

Lebesgue integral of $\chi_{\mathbb{Q}}: \mathbb{R} \rightarrow \mathbb{R}$

Problem involving decomposition of measures

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