List questions from lp-spaces

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An element of $ l^p $ may be regarded as a complex sequence.

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Give an example of a $f$ such that $\mathcal{L}^p(\mathbb{R})\not\ni f\in\mathcal{L}^q(\mathbb{R})$ $(1\leq p\leq q)$

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Hilbert transform of $L^{2} (- \pi, \pi)$ functions

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Finding bounded linear functionals on $L^{\infty}(\mathbb{R})$

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Is it true that if for $\varphi(p):=\int_E|f|^pd\mu$ we have $l=\lim_{x\uparrow b}\varphi(x)<+\infty$, then $\varphi(b)=l$?

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Sequence in $L^p$

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Intersection of $L^{p}$ spaces on two sets

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Sequences in $L^p$ and Orlicz spaces

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Proving completeness of $L_{\infty} space$

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Frechet differential in $L^\infty$ spaces

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Sequence of functions in $L^p \cap L^q$

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Convergence of supremum of the series of a signed measure

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Every order interval in $l^1$ is norm compact

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Exercise chapter 2.12 real analysis by E.M. Stein and R.Shakarchi

Published on 16 Apr 2026 - 14:16

List Question

Convolution inequality with weak Lp

An element of $ l^p $ may be regarded as a complex sequence.

Give an example of a $f$ such that $\mathcal{L}^p(\mathbb{R})\not\ni f\in\mathcal{L}^q(\mathbb{R})$ $(1\leq p\leq q)$

Hilbert transform of $L^{2} (- \pi, \pi)$ functions

Finding bounded linear functionals on $L^{\infty}(\mathbb{R})$

Is it true that if for $\varphi(p):=\int_E|f|^pd\mu$ we have $l=\lim_{x\uparrow b}\varphi(x)<+\infty$, then $\varphi(b)=l$?

Sequence in $L^p$

Intersection of $L^{p}$ spaces on two sets

Sequences in $L^p$ and Orlicz spaces

Proving completeness of $L_{\infty} space$

Frechet differential in $L^\infty$ spaces

Sequence of functions in $L^p \cap L^q$

Convergence of supremum of the series of a signed measure

Every order interval in $l^1$ is norm compact

Exercise chapter 2.12 real analysis by E.M. Stein and R.Shakarchi

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