Set of extreme points in $\ell^1(2) \oplus_2 \mathbb{R}.$

Question

Let $\ell^\infty(2)$ be a subset of $\mathbb{R}^2$ with maximum norm, that is, $\|(x,y)\|_\infty=\max\{|x|,|y| \}.$ Given two Banach spaces $X$ and $Y,$ if we have $X\oplus_2Y,$ we mean that $(x,y)\in X \oplus_2 Y$ with norm $\|(x,y\|_2= \sqrt{\|x\|_X^2 + \|y\|_Y^2}.$

We say that $x^*$ is an extreme point of unit ball $B_{X^*}$ of $X^*$ if whenever $x^*=\lambda y^* +(1-\lambda)z^*$ for some $\lambda\in[0,1]$ and $y^*,z^*\in B_{X^*},$ then $x^* = y^*$ or $x^*=z^*.$ Geometrically speaking, extreme points are 'corners' of unit ball.

Given a Banach space $X$, its set of extreme points $Ext(X)$ is defined to be a subset of closed unit ball of its dual $X^*.$ Also, if $X$ and $Y$ are isometrically isomorphic, we denote it as $X\cong Y.$

Question: Let $E = \ell^\infty(2)\oplus_2 \mathbb{R}$ be given. What is its $Ext(E)?$

First, I guess that $E^* \cong E$ (not very sure) due to $(\ell^2)^* \cong \ell^2$ and $F^* \cong F$ for any finite dimensional space $F$. So $Ext(E) \subseteq E^* \cong E.$

However, I have no idea how $E^*$ look like, thus cannot see its corner points. Any hint would be appreciated.

UPDATED: Based on Daniel Fischer's comment, dual space of $E$ is $\ell^1(2) \oplus_2 \mathbb{R}.$

So my question is reduced to finding extreme points of closed unit ball in $\ell^1(2) \oplus_2 \mathbb{R}.$

Answer 1

Some points to make first:

1. Extreme points are not preserved by isomorphisms, so thinking of what the given space is isomorphic to is not helpful. They are preserved by isometric isomorphisms.
2. A space is strictly convex if $\|a+b\|<\|a\|+\|b\|$ whenever $a,b$ are non-collinear vectors. In a strictly convex space, every boundary point of the unit ball is its extreme point.
3. If $X$ and $Y$ are strictly convex, then $X\oplus_2Y$ is also strictly convex. Indeed, to have equality $$\|(x_1,y_1) + (x_2,y_2)\| = \|(x_1,y_1)\| + \|(x_2,y_2)\|$$ one needs $x_1,x_2$ to be collinear, $y_1,y_2$ to be collinear, and also the Minkowski inequality $$ \sqrt{(\|x_1\|+\|x_2\|)^2 + (\|y_1\|+\|y_2\|)^2} \le \sqrt{\|x_1\|^2 + \|y_1\|^2} + \sqrt{\|x_2\|^2 + \|y_2\|^2} $$ must turn into equality. All this together implies that the vectors $(x_1,y_1)$ and $(x_2,y_2)$ are collinear.

From 1-3 the solution of the question about $\ell^\infty(2)\oplus_2 \mathbb{R}$ follows.

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Given a Banach space $X$, its set of extreme points $\operatorname{Ext}(X)$ is defined to be a subset of closed unit ball of its dual $X^∗$

It's not defined like that, as pointed out in comments.