Banach spaces with simple best approximate solutions

2012-01-17T19:25:52Z

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?

Banach spaces with simple best approximate solutions - MathOverflow

Banach spaces with simple best approximate solutions