Are finite-dimensional Banach spaces superreflexive?

Question

I'm wondering if every finite-dimensional Banach space is superreflexive. I've tried it via the definition by I got stuck. Is there a property or a result that I may have in consideration? Or does this property not hold for every finite-dimensional Banach space?

Thank you for helping!!

Answer 1

A Banach space is super-reflexive iff it is uniformly convex for an equivalent norm (Theodore W. Palmer, Banach Algebras and the General Theory of Star-algebras, p.82-83). Since every Hilbert space is uniformly convex, every finite-dimensional Banach space is super-reflexive.

Answer 2

A Banach space $X$ is super-reflexive if every Banach space $Y$ finitely representable in $X$ is reflexive. Now if $X$ is finite-dimensional and $Y$ is finitely representable in $X$, $Y$ must also be finite-dimensional with dimension at most the dimension of $X$, because every finite-dimensional subspace of $Y$ must be isomorphic to a subspace of $X$ and therefore has dimension at most that of $X$. But every finite-dimensional Banach space is reflexive. Therefore every finite-dimensional Banach space is super-reflexive.