Introduction

Abstract

Every mathematician working in Banach space geometry or Approximation theory knows, from his own experience, that most “natural” geometric properties may fail to hold in a general normed space unless the space is an inner product space. To recall the well known definitions, this means $\left\| x \right\| = < x,x{ > ^{{\raise0.5ex\hbox{$\scriptstyle 1$} \kern-0.1em/\kern-0.15em \lower0.25ex\hbox{$\scriptstyle 2$}}}}$ , where <x,y> is an inner (or: scalar) product on E, i.e. a function from E×E to the underlying (real or complex) field satisfying:

• (i) <x,x> 0 for x ≠ 0.

• (ii) <x,y> is linear in x.

• (iii) $< x,y > = < \overline {y,x} >$ (in the real case, this is just <x,y> = <y,x>).