Limit of operator norm Let $n$ be an integer. Let $T : \mathbb{R}^n \to \mathbb{R}^n$ be a linear map.
How to show that
$$
\lim_{p \to 1} \|T: \ell^1_n \to \ell^p_n\|
=\|T: \ell^1_n \to \ell^1_n\| ?
$$
Any reference is welcome.
 A: Indeed the key point is that $p$-norm is a continuous function as stated in the comments. Although, a small argument is needed:
Fix $x=(x_1,...,x_n)\in \mathbb{R^n}$ (i assume that your $p$ is $>1$). Then, by Hölder's Inequality used in the vectors $x,y\in \mathbb{R^n}$ where $y=(1,...,1)$ we have
\begin{align}
\sum_{k=1}^{n}|x_k|&=\sum_{k=1}^{n}|x_k|\cdot 1\\
&\leq \biggl(\sum_{k=1}^{n}|x_k|^p\biggr)^{1/p}\biggl(\sum_{k=1}^n1\biggr)^{1-1/p}\\
&=n^{1-1/p}\cdot ||x||_p
\end{align}
Hence,
$$\tag{1} ||x||_1\leq n^{1-1/p}||x||_p$$
Now, denote by $||T||_{1,1}$ the norm of the operator $T:l_1\to l_1$ and by $||T||_{1,p}$ the norm of the operator $T:l_1\to l_p$. Using $(1)$ we have
$$||Tx||_1\leq n^{1-1/p}\cdot ||Tx||_p\leq n^{1-1/p}\cdot ||T||_{1,p}\cdot ||x||_1$$
Hence, $||T||_{1,1}\leq n^{1-1/p}||T||_{1,p}$ so, taking $\liminf_{p\to 1}$ in the last inequality we get
$$||T||_{1,1}\leq \liminf_{p\to 1}||T||_{1,p}$$
For the other direction we use the continuity of the $p-$norm. Let $\epsilon>0$, by the definition of $||T||_{1,p}$ there is an $||x||_1\leq 1$ such that
$$\tag{2} ||T||_{1,p}-\epsilon<||Tx||_p\leq ||T||_{1,p}$$
Then, using $\lim_{p\to 1}||Tx||_p=||Tx||_1$ in $(2)$ we get $$\tag{3}\limsup_{p\to 1}||T||_{1,p}-\epsilon\leq ||Tx||_1\leq ||T||_{1,1}\cdot ||x||_1=||T||_{1,1}$$
Hence, $\limsup_{p\to 1}||T||_{1,p}\leq ||T||_{1,1}$. Putting all together,
$$\limsup_{p\to 1}||T||_{1,p}\leq ||T||_{1,1}\leq \liminf_{p\to 1}||T||_{1,p}$$
which is equivalent to $\lim_{p\to 1}||T||_{1,p}=||T||_{1,1}$.
