Let's say you're wondering around P(oo) (which I'll use to represent the space of polymomials of any degree on the interval [-1,1]) and you decide to calculate the matrix X representing the position operator x. Let's say you do this in the basis: 1, t, t^2, ..., t^n,... you'll find that the matrix X is not hermitian even though x is hermitian on P(oo). Then you decide to calculate the matrix again, but now you do it with the orthonormal basis generated from the above basis via the Gram-schmidt method (i.e. the properly normalized Legendre polynomials) Now you find that X is hermitian. What gives? should the hermiticity of an operator be independent of its representation in a particular basis? (maybe not and this is super obvious, feel free in that case to slap me).