Question

Construct a sequence of continuous functions $f_n: \mathbb{R} \rightarrow \mathbb{R} $such that $f_n$ converges to $f$ in $L_1 ({\lambda})$.

score 103 · Answer 1 · 2026-04-15 22:49:17

103

Views

Construct a sequence of continuous functions $f_n: \mathbb{R} \rightarrow \mathbb{R} $such that $f_n$ converges to $f$ in $L_1 ({\lambda})$.

Published on 15 Apr 2026 - 22:49

score 43 · Answer 2 · 2026-04-16 18:36:11

43

Views

Let $g_n(x)=\frac{1}{x} \chi_{(1,n)} (x) : \mathbb{R} \rightarrow \mathbb{R}$. Show $\forall$ $n>1$, $\exists$ a m>1 such that $||g_n - g_m||_1>1$.

Published on 16 Apr 2026 - 18:36

score 113 · Answer 3 · 2026-04-16 09:30:39

113

Views

Show $\lim_{|a|\to\infty}\int_a^{a+\epsilon} f(u(x)) d x = 0$

Published on 16 Apr 2026 - 9:30

score 83 · Answer 4 · 2026-04-14 20:52:18

83

Views

I am unsure what this question is asking - bounded linear functionals!

Published on 14 Apr 2026 - 20:52

score 1.9k · Answer 5 · 2026-04-14 12:53:18

1.9k

Views

Strong convexity of squared $\ell_p$ norm in Bregman divergence

Published on 14 Apr 2026 - 12:53

score 233 · Answer 6 · 2026-04-16 15:56:01

233

Views

If $\phi: X \rightarrow \mathbb{R} $ is a bounded linear functional, then the image $\phi(X)$ of $\phi$ is either $\{ 0\}$ or $\mathbb{R}$

Published on 16 Apr 2026 - 15:56

score 136 · Answer 7 · 2026-04-14 17:25:30

136

Views

For which $\alpha$ does this function belongs to $L_p(\mathbb{R}^n)$

Published on 14 Apr 2026 - 17:25

score 1.1k · Answer 8 · 2026-04-14 15:37:29

1.1k

Views

Kolmogorov - M. Riesz - Fréchet Theorem

Published on 14 Apr 2026 - 15:37

score 45.1k · Answer 9 · 2026-04-14 05:50:14

45.1k

Views

Why is every $p$-norm convex?

Published on 14 Apr 2026 - 5:50

score 63 · Answer 10 · 2026-04-15 22:27:25

63

Views

Prove characterization of the finiteness of $E \left[ |X| \right]$ by the behaviour of expectations $E \left[ |X| \mathbb{1}_{A}\right]$

Published on 15 Apr 2026 - 22:27

score 47 · Answer 11 · 2026-04-14 12:32:52

47

Views

sequence of functions not in $L^p$ converging to a function in $L^p$

Published on 14 Apr 2026 - 12:32

score 68 · Answer 12 · 2026-03-25 11:16:33

68

Views

Prove that the spectrum of an operator on L^p (\mathbb{R}) is the boundary of unit cỉrcle

Published on 25 Mar 2026 - 11:16

score 267 · Answer 13 · 2026-04-15 18:12:23

267

Views

Volume ratio of $\ell_1$ balls and $\ell_1$ surfaces

Published on 15 Apr 2026 - 18:12

score 211 · Answer 14 · 2026-04-15 14:43:35

211

Views

$f \in L^1(R)$, Fourier transform $\hat{f} \in L^2(R)$ implies $f \in L^2(R)$

Published on 15 Apr 2026 - 14:43

score 55 · Answer 15 · 2026-04-25 16:54:22

55

Views

Is this a Sobolev function?

Published on 25 Apr 2026 - 16:54

List Question

Construct a sequence of continuous functions $f_n: \mathbb{R} \rightarrow \mathbb{R} $such that $f_n$ converges to $f$ in $L_1 ({\lambda})$.

Let $g_n(x)=\frac{1}{x} \chi_{(1,n)} (x) : \mathbb{R} \rightarrow \mathbb{R}$. Show $\forall$ $n>1$, $\exists$ a m>1 such that $||g_n - g_m||_1>1$.

Show $\lim_{|a|\to\infty}\int_a^{a+\epsilon} f(u(x)) d x = 0$

I am unsure what this question is asking - bounded linear functionals!

Strong convexity of squared $\ell_p$ norm in Bregman divergence

If $\phi: X \rightarrow \mathbb{R} $ is a bounded linear functional, then the image $\phi(X)$ of $\phi$ is either $\{ 0\}$ or $\mathbb{R}$

For which $\alpha$ does this function belongs to $L_p(\mathbb{R}^n)$

Kolmogorov - M. Riesz - Fréchet Theorem

Why is every $p$-norm convex?

Prove characterization of the finiteness of $E \left[ |X| \right]$ by the behaviour of expectations $E \left[ |X| \mathbb{1}_{A}\right]$

sequence of functions not in $L^p$ converging to a function in $L^p$

Prove that the spectrum of an operator on L^p (\mathbb{R}) is the boundary of unit cỉrcle

Volume ratio of $\ell_1$ balls and $\ell_1$ surfaces

$f \in L^1(R)$, Fourier transform $\hat{f} \in L^2(R)$ implies $f \in L^2(R)$

Is this a Sobolev function?

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