I'm trying to understand how one can define the Ricci flow equation.

First you have to parametrized the set of all Riemannian metrics. Then you have to define the derivative on this parametrized family.

HERE IS MY PROBLEM.

The set of all metrics has the structure of a vector space (2-tensor fields). But this is not enough, We need to be this a normed vector space.

Is the set of all Riemannian metrics a Normed vector space? If not, How we can define the derivative ?

merelyan ODE in an infinite dimensional vector space. The equations the comes from a PDE are more special and some generalities one has to deal with when dealing with infinite dimensional ODEs can be avoided. $\endgroup$ – Willie Wong Apr 30 '15 at 11:49