Question

Suppose $f_n\to f$ in $L^1([0,1],\lambda)$. Prove or disprove: $\exists \{f_{n_j}\}$ such that $f_{n_j}(x)\to f(x)$ for almost every $x\in[0,1]$.

score 460 · Answer 1 · 2026-04-17 08:24:09

460

Views

Suppose $f_n\to f$ in $L^1([0,1],\lambda)$. Prove or disprove: $\exists \{f_{n_j}\}$ such that $f_{n_j}(x)\to f(x)$ for almost every $x\in[0,1]$.

Published on 17 Apr 2026 - 8:24

score 43 · Answer 2 · 2026-04-12 09:13:53

43

Views

Is there any relationship between $\mathcal{l}^p$ spaces and $\mathcal{L}^P$ space?

Published on 12 Apr 2026 - 9:13

score 274 · Answer 3 · 2026-04-17 06:49:41

274

Views

If $f_n$ converges to $f$ in $L^1(\mathbb{R})$ and $f_n$ converges to $g$, what relation exists between $f$ and $g$?

Published on 17 Apr 2026 - 6:49

score 31 · Answer 4 · 2026-04-16 08:50:08

31

Views

$L^1$ approximation by a slightly "displaced" copy

Published on 16 Apr 2026 - 8:50

score 288 · Answer 5 · 2026-04-12 00:33:08

288

Views

Upper and lower bound on $L^1$ norm purely in terms of measure

Published on 12 Apr 2026 - 0:33

score 39 · Answer 6 · 2026-03-29 03:36:26

39

Views

Writing an operator $T$ defined by $(T f)(t) = \int_{-\pi}^\pi h(t − s)f(s)ds$ as $\sum_{n \in \mathbb Z} \mu_n \langle f, \varphi_n\rangle \varphi_n$

Published on 29 Mar 2026 - 3:36

score 726 · Answer 7 · 2026-04-11 00:12:44

726

Views

$\ell^1$ is not complete for the norm $\|\cdot\|_\infty$

Published on 11 Apr 2026 - 0:12

score 746 · Answer 8 · 2026-04-14 12:57:06

746

Views

Supremum of absolute value of the Fourier transform equals $1$, and it is attained exactly at $0$

Published on 14 Apr 2026 - 12:57

score 102 · Answer 9 · 2026-04-16 23:25:13

102

Views

Question on Inequality from Bartle's Elements of Integration: Riesz Fischer Theorem

Published on 16 Apr 2026 - 23:25

score 248 · Answer 10 · 2026-04-11 06:10:31

248

Views

Why are there no finitely additive measures on $\ell_\infty$ for which the measure of every ball is positive and finite?

Published on 11 Apr 2026 - 6:10

score 659 · Answer 11 · 2026-04-14 19:28:58

659

Views

Proof verification: Something similar to Riesz-Fischer Theorem

Published on 14 Apr 2026 - 19:28

score 180 · Answer 12 · 2026-04-13 13:43:05

180

Views

Why does Rudin say that $f$ is in $L^\infty$?

Published on 13 Apr 2026 - 13:43

score 39 · Answer 13 · 2026-04-11 15:35:21

39

Views

Show linearity of a functional if it holds for nonnegatives

Published on 11 Apr 2026 - 15:35

score 285 · Answer 14 · 2026-04-19 04:05:01

285

Views

Explanation of spaces of functions in PDE

Published on 19 Apr 2026 - 4:05

score 61 · Answer 15 · 2026-04-16 00:41:38

61

Views

Characterization of measures such that $\frac{1}{x} \in L^1(H)$

Published on 16 Apr 2026 - 0:41

List Question

Suppose $f_n\to f$ in $L^1([0,1],\lambda)$. Prove or disprove: $\exists \{f_{n_j}\}$ such that $f_{n_j}(x)\to f(x)$ for almost every $x\in[0,1]$.

Is there any relationship between $\mathcal{l}^p$ spaces and $\mathcal{L}^P$ space?

If $f_n$ converges to $f$ in $L^1(\mathbb{R})$ and $f_n$ converges to $g$, what relation exists between $f$ and $g$?

$L^1$ approximation by a slightly "displaced" copy

Upper and lower bound on $L^1$ norm purely in terms of measure

Writing an operator $T$ defined by $(T f)(t) = \int_{-\pi}^\pi h(t − s)f(s)ds$ as $\sum_{n \in \mathbb Z} \mu_n \langle f, \varphi_n\rangle \varphi_n$

$\ell^1$ is not complete for the norm $\|\cdot\|_\infty$

Supremum of absolute value of the Fourier transform equals $1$, and it is attained exactly at $0$

Question on Inequality from Bartle's Elements of Integration: Riesz Fischer Theorem

Why are there no finitely additive measures on $\ell_\infty$ for which the measure of every ball is positive and finite?

Proof verification: Something similar to Riesz-Fischer Theorem

Why does Rudin say that $f$ is in $L^\infty$?

Show linearity of a functional if it holds for nonnegatives

Explanation of spaces of functions in PDE

Characterization of measures such that $\frac{1}{x} \in L^1(H)$

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