Question

Hints on proving $\liminf_{s\to 0}\sup_{f\in L^2(X):||f||_{L^2}\leq 1}\int_X|f(x)|^2|e^{-is/||x||}-1|^2dx>0$ for $X=\mathbb{R}^n$

score 35 · Answer 1 · 2026-03-30 00:22:43

35

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Hints on proving $\liminf_{s\to 0}\sup_{f\in L^2(X):||f||_{L^2}\leq 1}\int_X|f(x)|^2|e^{-is/||x||}-1|^2dx>0$ for $X=\mathbb{R}^n$

Published on 30 Mar 2026 - 0:22

score 161 · Answer 2 · 2026-04-03 12:35:54

161

Views

Showing that $\{\sqrt{2}\sin(\pi n x): n\in\mathbb{N}_0\}$ is an orthonormal basis for the space $L^2([0,1])$ given that $e^{2\pi il x}$ forms a basis

Published on 03 Apr 2026 - 12:35

score 67 · Answer 3 · 2026-03-27 18:14:58

67

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How to show that the sequence $\{T (e_n)\}_{n \geq 1}$ is square summable?

Published on 27 Mar 2026 - 18:14

score 46 · Answer 4 · 2023-04-18 15:43:56

46

Views

How to apply Gaussian Poincaré inequality

Published on 18 Apr 2023 - 15:43

score 56 · Answer 5 · 2026-03-30 10:40:10

56

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Evaluation of $L^2$ norm of exponential function on unit disk $\|\exp{z}\|_{L^2}$.

Published on 30 Mar 2026 - 10:40

score 52 · Answer 6 · 2023-04-22 10:11:30

52

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Meaning of $L_t^\infty L^2(dx d\gamma)$ and $L_t^2 L^2(dx d\gamma)$

Published on 22 Apr 2023 - 10:11

score 72 · Answer 7 · 2023-04-22 13:47:33

72

Views

Proving that $C_{c}(R)$ is not dense in $L^{\infty}(R)$

Published on 22 Apr 2023 - 13:47

score 76 · Answer 8 · 2023-04-23 12:11:14

76

Views

Find a function $\psi$ s.t. $\forall f\in\mathcal{F},f\le\psi$ a.e. , where $\mathcal{F}$ is a given subset in $L^{1}(\mathbb{R})$

Published on 23 Apr 2023 - 12:11

score 66 · Answer 9 · 2026-02-23 22:52:18

66

Views

Proof that bilinear form $(u,v)_{L^2(\Omega)} \mapsto (\nabla u, \nabla v)_{L^2(\Omega)}$ is not coercive in $H^1(\Omega)$

Published on 23 Feb 2026 - 22:52

score 42 · Answer 10 · 2023-04-25 15:34:05

42

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$\sum_{x\in X}|f(x)|=\sup\left\{\sum_{x\in F}|f(x)|:F \subset X\text{ is finite set} \right\}$

Published on 25 Apr 2023 - 15:34

score 151 · Answer 11 · 2023-04-26 09:41:20

151

Views

Prove that any $L^p$ function can be approximated with a sequence converging weakly but not strongly

Published on 26 Apr 2023 - 9:41

score 44 · Answer 12 · 2026-03-30 06:50:00

44

Views

Prove $\sup_{B_{1/2}(0)} |D^k u| \leq C \| u \|_{L^2(B_1(0))}$, and why do we have $L^2$ norm

Published on 30 Mar 2026 - 6:50

score 88 · Answer 13 · 2023-04-29 20:33:18

88

Views

Determine the orthogonal projection of $f$ onto $K := \{u \in H : |u| \le h \text{ a.e.}\}$

Published on 29 Apr 2023 - 20:33

score 95 · Answer 14 · 2023-05-02 11:07:58

95

Views

Example of a sequence in the closed unit ball of $\ell^1$ which does not have any weakly convergent subsequence.

Published on 02 May 2023 - 11:07

score 394 · Answer 15 · 2023-05-04 16:10:49

394

Views

L2 and L1 norm inverse relationship for bounded functions

Published on 04 May 2023 - 16:10

List Question

Hints on proving $\liminf_{s\to 0}\sup_{f\in L^2(X):||f||_{L^2}\leq 1}\int_X|f(x)|^2|e^{-is/||x||}-1|^2dx>0$ for $X=\mathbb{R}^n$

Showing that $\{\sqrt{2}\sin(\pi n x): n\in\mathbb{N}_0\}$ is an orthonormal basis for the space $L^2([0,1])$ given that $e^{2\pi il x}$ forms a basis

How to show that the sequence $\{T (e_n)\}_{n \geq 1}$ is square summable?

How to apply Gaussian Poincaré inequality

Evaluation of $L^2$ norm of exponential function on unit disk $\|\exp{z}\|_{L^2}$.

Meaning of $L_t^\infty L^2(dx d\gamma)$ and $L_t^2 L^2(dx d\gamma)$

Proving that $C_{c}(R)$ is not dense in $L^{\infty}(R)$

Find a function $\psi$ s.t. $\forall f\in\mathcal{F},f\le\psi$ a.e. , where $\mathcal{F}$ is a given subset in $L^{1}(\mathbb{R})$

Proof that bilinear form $(u,v)_{L^2(\Omega)} \mapsto (\nabla u, \nabla v)_{L^2(\Omega)}$ is not coercive in $H^1(\Omega)$

$\sum_{x\in X}|f(x)|=\sup\left\{\sum_{x\in F}|f(x)|:F \subset X\text{ is finite set} \right\}$

Prove that any $L^p$ function can be approximated with a sequence converging weakly but not strongly

Prove $\sup_{B_{1/2}(0)} |D^k u| \leq C \| u \|_{L^2(B_1(0))}$, and why do we have $L^2$ norm

Determine the orthogonal projection of $f$ onto $K := \{u \in H : |u| \le h \text{ a.e.}\}$

Example of a sequence in the closed unit ball of $\ell^1$ which does not have any weakly convergent subsequence.

L2 and L1 norm inverse relationship for bounded functions

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