Let $c$ denote the space of convergent sequences in $\mathbb C$, $c_0\subset c$ be the space of all sequences that converge to $0$. Given the uniform metric, both of them can be made into Banach spaces. It can be shown that the dual spaces of them are isometrically isomorphic, i.e. $c^*\cong c_0^*$. Are $c$ and $c_0$ isometrically isomorphic? If not, how can one show the absence of such a isometric isomorphism? Thanks!
Functional Analysis – Are These Two Banach Spaces Isometrically Isomorphic?
banach-spacesfunctional-analysis
Best Answer
The closed unit ball of $c_0$ has no extreme points. The closed unit ball of $c$ has many extreme points, such as $(1,1,\ldots)$. Since the property of being an extreme point is preserved by isometries, $c$ and $c_0$ are not isometrically isomorphic.