  We address the question about the reasons why the "Wick-rotated",
positive-definite, space-time metric obeys the Pythagorean theorem. An answer
is proposed based on the convexity and smoothness properties of the functional
spaces purporting to provide the kinematic framework of approaches to quantum
gravity. We employ moduli of convexity and smoothness which are eventually
extremized by Hilbert spaces. We point out the potential physical significance
that functional analytical dualities play in this framework. Following the
spirit of the variational principles employed in classical and quantum Physics,
such Hilbert spaces dominate in a generalized functional integral approach. The
metric of space-time is induced by the inner product of such Hilbert spaces.
