# Questions tagged [lp-spaces]

For questions about $L^p$ spaces, that is, given a measure space $(X,\mathcal F,\mu)$, the vector space of equivalence class of measurable functions with $p$-th power of the absolute value integrable. Question can be about properties of elements of these spaces, or when the ambient space on a problem is a $L^p$ space.

3,125 questions
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### Existence of functionals on $L^0$

Studying a paper about risk measures by F. Delbaen, I came into this statement: Let $(\Omega,\mathcal{F},\mathbb{P})$ be a probability space: if $\mathbb{P}$ is atomless, then there exists no ...
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### Are $c_0$ and $c$ duals of some spaces?

The (continuous) dual of a normed vector space is always a Banach space, but the converse is not true. That is, not all Banach spaces are isomorphic to the dual space of some normed vector space. ...
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### Does $\int^{\infty}_{0}pa^{p-1}1_{\{|f|>a\}}(x)da=\int^{|f(x)|}_{0}pa^{p-1}da$?

Let $f\in L^p(\mathbb{R}^d)$ for $p\in [1,\infty)$. Show that $$\|f\|_p=\left(\int^{\infty}_{0}pa^{p-1}m\{|f|>a\} \right)^{1/p}$$ My attempt: We can use Fubini's theorem in the following way: \...
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### Construction of oscillating sequence in $\mathcal{L}^{\infty}(\Omega, \{z_1, …, z_N\})$

Let $N \in \mathbb{N}$, $\lambda_n \in (0,1)$ , $z_n \in \mathbb{R}^N$ for $n \leq N$ such that \begin{equation} \sum_{n=1}^N \lambda_n = 1,\quad \sum_{n=1}^N \lambda_n z_n = z \in \mathbb{R}^N. \...
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### Bounded Sequence in $L^\infty$ and Interpolation in $L^p$
a) Let $1\leq p_1\leq p\leq p_2\leq \infty$ and for $\alpha \in [0,1]$ $\frac {1}{p}=\frac {\alpha}{p_1}+\frac {1- \alpha}{p_2}$ Prove that if $f\in L^{p_1}\cap L^{p_2}$, then $f\in L^p$ and we have ...
### Verification of alternative proof of $\lim_{p\to \infty}\|u\|_p=\|u\|_\infty$
I have to show that $$\lim_{p\to \infty}\|u\|_p=\|u\|_\infty$$ Suppose $u\in L^\infty (E)$ for measurable $E \subset \mathbb{R}^d$ having finite measure. I come up with this proof: \Big| \|u\|_p -...