Let us define $\|.\|$ on $\ell^2$ as follows: $$\|x\|=\max\begin{Bmatrix}2(\sum\limits_{n=2}^{\infty}x_n^2)^{\frac{1}{2} }, (\sum\limits_{n=1}^{\infty}x_n^2)^{\frac{1}{2}}\end{Bmatrix} \text{ for all }x\in \ell^2.$$
I want to check whether $(\ell^2,\|.\|)$ is smooth. My attempt is as follows:
Any two dimensional vector subspace of $\ell^2$ is isomorphic to $\mathbb{R}^2$. The question is to determine the norm. Here two cases come up for consideration.
Case I: $\|(x,y)\|=\max\{2|y|, \sqrt{x^2+y^2}\}$ for all $(x,y)\in \mathbb{R}^2$ and
Case II: $\|(x,y)\|=\max\{2\sqrt{x^2+y^2}, \sqrt{x^2+y^2}\}$ for all $(x,y)\in \mathbb{R}^2$.
In Case I, the closed unit ball is the intersection of $\{(x,y)\in \mathbb{R}^2: x^2+y^2\leq 1\}$ and $\{(x,y)\in \mathbb{R}^2:|y|\leq \frac{1}{2}\}$, and in the second case it is the set $\{(x,y)\in \mathbb{R}^2: x^2+y^2\leq \frac{1}{4}\}$. In both the cases every point in the closed unit ball can not be supported by more than one linear functional. Thus $(\mathbb{R}^2,\|.\|)$ is smooth.
Now, we know that a normed space is smooth if and only if each of its two dimensional subspaces is smooth. Hence $(\ell^2,\|.\|)$ is smooth. Is my attempt correct or I have committed mistake in considering two dimensional subspaces? Any help is appreciated.