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!
What a remarkably silly thing to say! Of course all continuous functions (on a compact interval) are integrable. I must have been thinking of "differentiable".
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.