Let $1\le p<+\infty$ and let $T:l^p(\mathbb{N}) \rightarrow L^p([0,+\infty),dx)$ an application defined by
$T(a)(x)=\sum_{n=1}^{+\infty}a_n\mathbb{1}_{[n-1,n)}(x),\forall a=(a_n)_{n\in\mathbb{N}}\in l^p(\mathbb{N}),\forall x\in[0,+\infty)$.
Show that $T$ is an isometry.
I need to prove that
$||T(a)||_{L^p}=||a||_{l^p}$.
Let $X=[0,+\infty)$
$||T(a)||_{L^p}=(\int_X |T(a)(x)|^p)^{1/p}=(\int_X |\sum_{n=1}^{+\infty}a_n\mathbb{1}_{[n-1,n)}(x)|^p)^{1/p}$.
Can you help me ?
So you are stuck somewhere wondering what is $|\sum_{n\geq 1} a_n\mathbf{1}_{[n-1,n)}(x)|^p$?
If $x_0\in X:=[0,\infty)$ then $x_0\in [n_0-1,n_0)$ for some $n_0\in\mathbb{N}$. Hence: $$\bigg|\sum_{n\geq 1} a_n\mathbf{1}_{[n-1,n)}(x_0)\bigg|^p=|a_{n_0}\mathbf{1}_{[n_0-1,n_0)}(x_0)|^p=|a_{n_0}|^p$$ We may conclude that $$\bigg|\sum_{n\geq 1} a_n\mathbf{1}_{[n-1,n)}(x)\bigg|^p=\sum_{n\geq 1}|a_n|^p \mathbf{1}_{[n-1,n)}(x)$$ Integrating and using The Monotone Convergence Theorem (MCT) yields: \begin{align} \Vert T(a)\Vert_{L^p}^p&=\int_X \bigg|\sum_{n\geq 1} a_n\mathbf{1}_{[n-1,n)}(x)\bigg|^p\,dx\\ &=\int_X\sum_{n\geq 1}|a_n|^p \mathbf{1}_{[n-1,n)}(x)\,dx\\ &\stackrel{MCT}{=}\sum_{n\geq 1 }\int_X|a_n|^p \mathbf{1}_{[n-1,n)}(x)\,dx\\ &=\sum_{n\geq 1} |a_n|^p\\ &=\Vert a\Vert_{l^p}^p \end{align} Taking $p$-th root gives the desired result.