Question

Proof of $\big|\int_E (f\cdot g)(x) \text{d}x\big| \leq [\int_E |f(x)|^p\text{d}x]^{(1/p)} \cdot [\int_E |g(x)| ^q\text{d}x]^{(1/q)}$

score 101 · Answer 1 · 2026-04-20 06:02:26

101

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Proof of $\big|\int_E (f\cdot g)(x) \text{d}x\big| \leq [\int_E |f(x)|^p\text{d}x]^{(1/p)} \cdot [\int_E |g(x)| ^q\text{d}x]^{(1/q)}$

Published on 20 Apr 2026 - 6:02

score 91 · Answer 2 · 2026-04-15 18:10:56

91

Views

Is the map $T: C(X) \to \mathbb{B}(L^2(X,\nu))$ defined by $T(f)=M_f$, where $M_f(g)=fg$, injective?

Published on 15 Apr 2026 - 18:10

score 25 · Answer 3 · 2026-04-14 12:14:13

25

Views

Is it true that $\|u^{a}\|_{L^\infty }=\|u\|_{L^\infty }^a$ where $a>0$?

Published on 14 Apr 2026 - 12:14

score 90 · Answer 4 · 2026-04-16 01:08:24

90

Views

If $f\in L^{1}((0,1))$, does it hold that $\lim\limits_{n\to+\infty}\int_{0}^{\frac{1}{n}}f(x)\mathrm{d}x=0$?

Published on 16 Apr 2026 - 1:08

score 54 · Answer 5 · 2026-04-16 11:00:50

54

Views

Is the set of continuous function that are also $\mathcal{L}^1(\mathbb{R})$ a subset of $\mathcal{L}^2(\mathbb{R})$

Published on 16 Apr 2026 - 11:00

score 71 · Answer 6 · 2026-03-26 02:11:44

71

Views

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score 1k · Answer 7 · 2026-04-29 12:52:50

1k

Views

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score 38 · Answer 8 · 2026-04-12 13:46:14

38

Views

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Published on 12 Apr 2026 - 13:46

score 31 · Answer 9 · 2026-04-29 12:46:39

31

Views

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score 2k · Answer 10 · 2026-04-09 21:35:39

2k

Views

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Published on 09 Apr 2026 - 21:35

score 459 · Answer 11 · 2026-05-04 11:34:03

459

Views

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Published on 04 May 2026 - 11:34

score 273 · Answer 12 · 2026-04-10 09:21:11

273

Views

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Published on 10 Apr 2026 - 9:21

score 88 · Answer 13 · 2026-04-12 18:15:51

88

Views

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Published on 12 Apr 2026 - 18:15

score 434 · Answer 14 · 2026-04-14 00:00:35

434

Views

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Published on 14 Apr 2026 - 0:00

score 108 · Answer 15 · 2026-04-09 03:01:27

108

Views

Convergence in norm equivalent formulations

Published on 09 Apr 2026 - 3:01

List Question

Proof of $\big|\int_E (f\cdot g)(x) \text{d}x\big| \leq [\int_E |f(x)|^p\text{d}x]^{(1/p)} \cdot [\int_E |g(x)| ^q\text{d}x]^{(1/q)}$

Is the map $T: C(X) \to \mathbb{B}(L^2(X,\nu))$ defined by $T(f)=M_f$, where $M_f(g)=fg$, injective?

Is it true that $\|u^{a}\|_{L^\infty }=\|u\|_{L^\infty }^a$ where $a>0$?

If $f\in L^{1}((0,1))$, does it hold that $\lim\limits_{n\to+\infty}\int_{0}^{\frac{1}{n}}f(x)\mathrm{d}x=0$?

Is the set of continuous function that are also $\mathcal{L}^1(\mathbb{R})$ a subset of $\mathcal{L}^2(\mathbb{R})$

A finite-codimensional subspace of the space of [square-integrable] holomorphic functions

Failing to prove that E(XY) is an inner product

Condition on $\nu$ which makes the inclusion map from $C(X)$ to $L^{\infty}(X)$ injective

When does an action on $X$ makes an well defined action on $L^{\infty}(X)$?

How are $L^P$ spaces defined for complex valued functions?

Equivalence between squared-norm and norm inequalities

Norms of higher order Lp spaces

Why is it nonsensical to evaluate $f(0)$ for $f\in L^p$?

Duality between $L^\infty$ and $L^1$

Convergence in norm equivalent formulations

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