Domain of Orthogonality of Legendre Polynomials? In my Numerical Analysis Course, under the topic Theory of Orthogonal Polynomials We are following the book Numerical Analysis by Kendall E Atkinson.
The problem is the author keeps saying "The Legendre Polynomials are orthogonal in $(-1,1)$"
According to me a polynomial is defined everywhere and is Orthogonal With respect to a Inner-product which might have some integral and limit.
It makes no sense for me to talk about orthogonality in a Domain.
The course is combined for Physics and Mathematics students and I am a Mathematics Major. 
Each Physics major believes the polynomial are orthogonal only in the domain.
for example Legendre polynomials are orthogonal in $(-1,1)$ only and you cant use them outside the $(-1,1)$ to study them or to find the coefficients using recursive relation outside $(-1,1)$ using Gaussian Elimination.
I want to know if there is meaning of Polynomials being orthogonal in a domain. If Yes, what does it mean?


"The Legendre Polynomials are orthogonal in (-1,1)"

The Legendre Polynomials are orthogonal with respect to the inner product $\langle f,g\rangle=\int_\mathbb R w(x)f(x) g(x)\, dx$ for a certain weight function $w(x).$
