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Consider a measure space $(S,\mu)$ and assume that $\mu(S)=1$. We consider the quantile function (or nonincreasing rearrangement) of a real valued function $f:S\to\mathbb{R}$ as the function \begin{equation} Hf(x)=\inf\{u>0:\mu(f>u)\leq x\}. \end{equation} For $w:[0,1]\to[1,\infty)$ a nonincreasing positive function, consider \begin{equation} \rho(f)=\int_0^1w(x)Hf(x)^p\mathrm{d}x \end{equation} and define $B$ as the collection of all $f:S\to \mathbb{R}$ such that $\rho(|f|)<\infty$.

Problem. It is not too difficult to show that $\|f\|=\rho(|f|)^{1/p}$ defines a (semi-)norm, which makes $(B,\|\cdot\|)$ a normed space. The question is whether this normed space is a Banach space?

To begin, the norm $\|\cdot\|$ satisfies the ideal property in that for any $|g|\leq |f|$, if $f\in B$, then $g\in B$ and $\|g\|\leq \|f\|$. According to the article Banach function spaces done right, if $(B,\|\cdot\|)$ has the so-called Fatou property, that is if $0\leq f_n\uparrow f$ for $(f_n)_{n\geq 1}$ in $B$ and $\sup_{n\geq 1}\|f_n\|<\infty$, then $f\in B$ and $\|f\|=\sup_{n\geq 1}\|f_n\|$, then $(B,\|\cdot\|)$ is complete. Yet, I seem to be stuck in actually providing an affirmative answer (or counter-example) to the question on whether $(B,\|\cdot\|)$ is a Banach space.

Any help is welcome!

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  • $\begingroup$ If $\|\cdot\|$ is only a seminorm, how is $(B,\|\cdot\|)$ a normed space? $\endgroup$ Commented Nov 11 at 16:46

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This was proven by Lorentz in ''Some new functional spaces'' (1950). The idea is the following.

Since $w(x) \geq 1$, we have the embedding $B \hookrightarrow{} L_p$. On the other hand, let $\epsilon > 0$ arbitrary, and define $\phi_{\epsilon}(f) = \big(\int_{\epsilon}^1 w(x)Hf(x)^p dx \big)^{1/p}$. Clearly, $\phi_{\epsilon}$ is continuous in $L_p$ and it is decreasing in $\epsilon$. Hence $\rho(f) = \lim_{\epsilon \to 0} \phi_{\epsilon}(f)$ is a lower semicontinuous function in $L_p$ (with values $0 \leq \rho \leq \infty$).

Now, consider a Cauchy sequence in $B$. Then, it is a Cauchy sequence in $L_p$, and by the completeness of $L_p$ we can find $f \in L_p$ such that $f_n \to f$ in $L_p$. Therefore, if we take $n_0 \in \mathbb{N}$ such that $\rho(|f_n - f_m|) < \epsilon$ for every $n,m \geq n_0$, we conclude by the lower semicontinuity of $\rho$ in $L_p$ that $$ \rho(|f_n - f|) \leq \liminf_{m \to \infty} \rho(|f_n - f_m|) < \epsilon, $$ for every $n \geq n_0$. Hence, $f \in B$ and $f_n \to f$ in $B$, as we wanted.

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