The "Gram-Schmidt orthogonalization process" is a method of changing any basis in a vector space (with inner product) to a an orthonormal basis- a basis such that the norm of each vector is 1 and two distinct vectors have inner product 0.
Whether you integrate "between 1 and 0" or from -1 to 1 or whether you
integrate at all depends upon what vector space you are talking about.
If you are talking about a set of continuous functions, you had better have some
other way of defining norms and inner products because
most continuous functions are not integrable at all!
It is common to define the "supremum norm" on continuous functions. If our vector space is the space of all functions continuous on [a, b] then we know that sup (f) (the least upper bound on |f(x)| for x in [a, b]) exists for every such f. But there is no "inner product" that gives that norm so "Gram- Schmidt" cannot be used for that set.
If you have the set of functions such that
is defined on [a, b], then we could define the norm as
(Often called "
") but, again, there is no "inner product" associated with that norm so "Gram-Schmidt" cannot be used.
The set of "square integrable functions" on a set,
, such that
exists, also has the property that
so that we can define the "inner product" of f and g in that way. That gives both an inner product and norm so that Gram-Schmidt
can be used.
But the problem is that "Gram-Schmidt" is applicable in much more examples than you seem to understand- so it is hard to know exactly what you are asking or what sort of response you would understand.