Question

A version of dominated convergence theorem

score 97 · Answer 1 · 2026-03-26 03:00:48

97

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A version of dominated convergence theorem

Published on 26 Mar 2026 - 3:00

score 53 · Answer 2 · 2026-03-26 07:35:31

53

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Let $a$ be continuous such that $|a(x)| \le C |x|^{p/q}$. Then $A:L^p(\Omega) \to L^q(\Omega), u \mapsto a \circ u$ is continuous

Published on 26 Mar 2026 - 7:35

score 85 · Answer 3 · 2026-03-25 11:04:38

85

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If $\mu(\Omega) < \infty$, then the space of $\mu$-simple functions is dense in $L^\infty (\Omega)$

Published on 25 Mar 2026 - 11:04

score 81 · Answer 4 · 2023-04-09 10:17:10

81

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Brezis' exercise 4.22.4: convergence in weak topology $\sigma(L^1, L^\infty)$

Published on 09 Apr 2023 - 10:17

score 32 · Answer 5 · 2026-03-26 03:11:05

32

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Prove that $C$ is closed in the weak$^*$ topology $\sigma(L^\infty, L^1)$

Published on 26 Mar 2026 - 3:11

score 53 · Answer 6 · 2026-04-07 08:12:18

53

Views

A refinement of dominated convergence theorem

Published on 07 Apr 2026 - 8:12

score 40 · Answer 7 · 2026-03-26 04:53:25

40

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Prove that $v_n \xrightarrow{n\to\infty} v$ in $L^\infty (\mathbb R^d)$ w.r.t. the weak$^*$ topology $\sigma(L^\infty, L^1)$

Published on 26 Mar 2026 - 4:53

score 33 · Answer 8 · 2026-03-26 06:34:47

33

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There is $(u_n) \subset \mathcal C_c^{\infty} (\Omega)$ such that $u_n \to u$ a.e. and $u_n \to u$ in the weak$^*$ topology $\sigma(L^\infty, L^1)$

Published on 26 Mar 2026 - 6:34

score 49 · Answer 9 · 2026-04-07 08:15:09

49

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A characterization of $f \in L^1 (\Omega)$ by duality

Published on 07 Apr 2026 - 8:15

score 70 · Answer 10 · 2026-04-07 08:17:47

70

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An equivalent condition of $f^+ \in L^1 (\Omega)$ for $f \in L^1_{\text{loc}} (\Omega)$

Published on 07 Apr 2026 - 8:17

score 110 · Answer 11 · 2026-04-07 08:11:22

110

Views

Approximate a function in $L^p (\mathbb R^d)$ by mollifiers

Published on 07 Apr 2026 - 8:11

score 48 · Answer 12 · 2026-04-07 08:17:46

48

Views

Let $\rho_n (x) := n^d \rho (nx)$. Then $\rho_n * f \xrightarrow{n \to \infty} f$ in $L^p$

Published on 07 Apr 2026 - 8:17

score 54 · Answer 13 · 2026-04-07 09:56:52

54

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Let $f \in \mathcal C (\mathbb R^d)$. Then $\rho_n * f \xrightarrow{n \to \infty} f$ uniformly on compact sets

Published on 07 Apr 2026 - 9:56

score 55 · Answer 14 · 2026-03-28 06:00:04

55

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The adjoint of $T_a: L^2 (\mathbb R^d) \to L^2 (\mathbb R^d), u \mapsto a * u$

Published on 28 Mar 2026 - 6:00

score 28 · Answer 15 · 2023-04-14 21:38:42

28

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Variant of fundamental lemma of calculus of variations

Published on 14 Apr 2023 - 21:38

List Question

A version of dominated convergence theorem

Let $a$ be continuous such that $|a(x)| \le C |x|^{p/q}$. Then $A:L^p(\Omega) \to L^q(\Omega), u \mapsto a \circ u$ is continuous

If $\mu(\Omega) < \infty$, then the space of $\mu$-simple functions is dense in $L^\infty (\Omega)$

Brezis' exercise 4.22.4: convergence in weak topology $\sigma(L^1, L^\infty)$

Prove that $C$ is closed in the weak$^*$ topology $\sigma(L^\infty, L^1)$

A refinement of dominated convergence theorem

Prove that $v_n \xrightarrow{n\to\infty} v$ in $L^\infty (\mathbb R^d)$ w.r.t. the weak$^*$ topology $\sigma(L^\infty, L^1)$

There is $(u_n) \subset \mathcal C_c^{\infty} (\Omega)$ such that $u_n \to u$ a.e. and $u_n \to u$ in the weak$^*$ topology $\sigma(L^\infty, L^1)$

A characterization of $f \in L^1 (\Omega)$ by duality

An equivalent condition of $f^+ \in L^1 (\Omega)$ for $f \in L^1_{\text{loc}} (\Omega)$

Approximate a function in $L^p (\mathbb R^d)$ by mollifiers

Let $\rho_n (x) := n^d \rho (nx)$. Then $\rho_n * f \xrightarrow{n \to \infty} f$ in $L^p$

Let $f \in \mathcal C (\mathbb R^d)$. Then $\rho_n * f \xrightarrow{n \to \infty} f$ uniformly on compact sets

The adjoint of $T_a: L^2 (\mathbb R^d) \to L^2 (\mathbb R^d), u \mapsto a * u$

Variant of fundamental lemma of calculus of variations

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