0
votes
1answer
31 views

norm on a quotient-space

Let $M:[0,\infty)\to[0,\infty)$ be continuous and convex. Further $M$ satisfies $M(t)=0\Leftrightarrow t=0$. Let $$\mathcal L_M(\mathbb R):=\left\{f:\mathbb R\to\mathbb R \mathrm{\ measurable\ ...
0
votes
1answer
64 views

proofread $\left \| f -g\right \|_{p}^{p}=p(p-1)\int_{0}^{\infty}\int_{0}^{t}[m(f>t/g>s)+ m(g>t/f>s)]|t-s|^{p-2}dsdt$

Is this argument correct? $\left \| f -g\right \|_{p}^{p}=$ $\int_{0}^{\infty}m(|f-g|>t^{\frac{1}{p}}) dt\stackrel{c.o.v.}{=}$ $p\int_{0}^{\infty}m(|f-g|>t) t^{p-1}dt\stackrel{t=|u-s|}{=}$ ...
0
votes
1answer
68 views

What is $L^1$ norm for $n$-fold functional space of integrable function?

Consider a space of $n$-fold real-valued integrable functions, $X \doteq [L^1([0,T])]^n$. Some use an alternative notation like $X \doteq L^1([0,T]; \mathbb{R}^n)$. Does $f\in X$ mean ...
2
votes
1answer
490 views

Inequality of Lebesgue integral with $L^p$-norm

Let $X_t(\omega)$ be a continuous function $t\rightarrow L^p(\omega)$ (i.e., if we fixed the variable $t$ we obtain a function which belongs to $L^p$), with $t\in[0,T]$ and $\omega\in\mathbb{R}$. I ...