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.