Setting initial values in MMA's Gram-Schmidt process

Question

How do you setup initial values in MMA's Gram-Schmidt process.

Take this example:

ClearAll[a, b, c, d, e, f]
Orthogonalize[{a - b, c - d, e - f}, Dot, Method -> "GramSchmidt"]

However, the first term in the result would, by definition, be a-b - this simplifies subsequent expressions.

Using the MathWorld notation, I'm asking how to setup the MMA calculation such that $\psi_0=0, \psi_1=a-b$.

I think the following is related, but happy to submit as a separate question:

I'd also like to be able to use a different notation for the inner product. Rather than the Dot notation (...).(...), is the following possible?:

\[DoubleStruckCapitalE][( ...) ( ...) | Subscript[\[ScriptCapitalF], 
  t - 1]]

Answer 1

Turns out you cannot do this using the built in function Orthogonalize.

Rather you need to meld answers from two previous questions: The first here. The second here.

Note:

This function is opinionated at inserts 0 as the first value returned in the list.

oneStepOrtogonalizeGen[vec_, {}, _, _, _] := vec;

oneStepOrtogonalizeGen[vec_, vecmat_List, dotF_, plusF_, timesF_] := 
  Fold[plusF[#1, timesF[-dotF[vec, #2]/dotF[#2, #2], #2]] &, vec, 
   vecmat];

GSOrthogonalizeGen[startvecs_List, dotF_, plusF_, timesF_] := 
  Module[{rr = Array[0 &, Length[startvecs] + 1], rtemp},
   rtemp = 
    oneStepOrtogonalizeGen[startvecs[[1]], {}, martingaleDot, 
     functionPlus, functionTimes];
   rr[[1]] = 0;
   rr[[2]] = rtemp;
   Do[
    rtemp = 
     oneStepOrtogonalizeGen[rtemp, {startvecs[[i]]}, martingaleDot, 
      functionPlus, functionTimes];
    rr[[i + 1]] = rtemp;
    , {i, 2, Length[startvecs]}];
   rr];

Now you can define your own inner product notation/evaluation, as well as set initial values as you like.