Let $\langle V,.\rangle$ be a Banach space such that:
$\;\;$ for all continuous linear maps $\: L : V\to V \:$ and members $v$ of $V$, there exists a unqiue member $u$ of $V$
$\;\;$ that minimizes $\langle L(u)+(v),u\rangle$ in the lexicographic order
$\;\;$ and
$\;\;\;$ for all continuous linear maps $\: L : V\to V \:$, $\:$ the function $\: L^{\dagger} : V\to V \:$ given by
$\;\;$ $\big[L^{\dagger}(v)$ minimizes $\langle L(L^{\dagger}(v))+(v),L^{\dagger}(v)\rangle$ in the lexicographic order$\big]$
$\;\;\;$ is continuous and linear
Does it follow that $\langle V,.\rangle$ is
$\;\;$ 1. $\:$ isometrically
$\;\;$ 2. $\:$ homeomorphically
isomorphic to a Hilbert space?
Take the 2minute tour
×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.


