orthogonality with a weight
Perpendicular is not an absolute notion for functions — it depends on how you measure angles, and that is what an inner product does. Orthogonality with a weight is orthogonality measured by a weighted inner product: instead of the plain integral of f times g, you use the integral of f times g times w, where w(x) is a fixed positive weight. Under this rule two functions are perpendicular when their weighted overlap is zero, even if their unweighted overlap is not.
The weighted inner product is defined as the pairing (f, g) = integral from a to b of f(x) g(x) w(x) dx, and the induced norm is the square root of (f, f). Every notion you know for ordinary vectors transfers verbatim: orthogonal means (f, g) = 0, the norm measures size, projecting f onto a unit-norm g returns (f, g) g, and Pythagoras holds. The whole point of working in self-adjoint Sturm-Liouville form is that it tells you exactly which w to use — the weight is the coefficient of lambda y in the equation, and it is the only weight for which the eigenfunctions of that problem come out orthogonal.
Why bother with a weight at all? Because real geometries and real physics put it there. In polar or spherical coordinates the area element drags in an r or a sin(theta) factor that becomes the weight; for a string of varying density the mass density is the weight. Carrying w through every integral keeps the orthogonal structure intact, so coefficient formulas and Parseval's identity stay clean. Drop the weight and the eigenfunctions stop being orthogonal, projection stops working, and the expansion silently breaks.
Chebyshev polynomials T_m, T_n are orthogonal on [-1, 1] only with the weight w(x) = 1/sqrt(1 - x^2): integral from -1 to 1 of T_m(x) T_n(x) / sqrt(1 - x^2) dx = 0 for m not equal to n. With the plain weight 1 they are not orthogonal.
Each polynomial family is orthogonal only under its own weight; switching the weight breaks orthogonality.
A common slip is to compute a coefficient as integral of f y_n dx without the weight; for a non-trivial weight (Bessel, Chebyshev, Hermite) that gives the wrong number and a series that does not reproduce f.