Question

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

score 53 · Answer 1 · 2026-03-26 07:35:31

53

Views

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 2 · 2026-03-25 11:04:38

85

Views

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 39 · Answer 3 · 2026-03-26 08:14:56

39

Views

countable intersection of domain of densely defined operators is dense

Published on 26 Mar 2026 - 8:14

score 32 · Answer 4 · 2026-03-26 03:11:05

32

Views

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

Published on 26 Mar 2026 - 3:11

score 52 · Answer 5 · 2023-04-09 22:48:18

52

Views

A refinement of dominated convergence theorem

Published on 09 Apr 2023 - 22:48

score 40 · Answer 6 · 2026-03-26 04:53:25

40

Views

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 48 · Answer 7 · 2023-04-10 19:55:55

48

Views

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

Published on 10 Apr 2023 - 19:55

score 51 · Answer 8 · 2026-03-28 05:28:44

51

Views

An element of $C(\mathbb{D})$ which is not in the connected component

Published on 28 Mar 2026 - 5:28

score 69 · Answer 9 · 2023-04-11 10:26:47

69

Views

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

Published on 11 Apr 2023 - 10:26

score 56 · Answer 10 · 2026-03-28 02:05:24

56

Views

Brezis' exercise 4.29: how to get rid of the dimension $d$?

Published on 28 Mar 2026 - 2:05

score 30 · Answer 11 · 2026-03-26 06:26:16

30

Views

distance beetween weakly closed sets

Published on 26 Mar 2026 - 6:26

score 86 · Answer 12 · 2026-03-28 07:08:08

86

Views

Example of a non zero multiplicative linear functional on a non commutative Banach algebra

Published on 28 Mar 2026 - 7:08

score 89 · Answer 13 · 2026-03-28 07:07:24

89

Views

Multiplicative linear functionals on B(H)

Published on 28 Mar 2026 - 7:07

score 109 · Answer 14 · 2023-04-11 16:09:01

109

Views

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

Published on 11 Apr 2023 - 16:09

score 47 · Answer 15 · 2023-04-11 16:37:49

47

Views

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

Published on 11 Apr 2023 - 16:37

List Question

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 $L^\infty (\Omega)$

countable intersection of domain of densely defined operators is dense

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 $\sigma(L^\infty, L^1)$

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

An element of $C(\mathbb{D})$ which is not in the connected component

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

Brezis' exercise 4.29: how to get rid of the dimension $d$?

distance beetween weakly closed sets

Example of a non zero multiplicative linear functional on a non commutative Banach algebra

Multiplicative linear functionals on B(H)

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 $L^p$

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