A satisfactory set of transformation operators

If an n-dimensional function space is defined by the set of linearly-independent basis functions. ../, . .., and fn and if these are 

The requirement that Ottfi produces a function belonging to the given function space (see eqn (5-6.2)) will be met by the proper choice of function space (see 5-8). If this is the case, however, we can write, for an n-dimensional function space defined by the linear ly-independent basis functions fx, ft. .., and fn, 

What is of supreme importance for us, is the fact that the nxn matrices D(P) in eqn (5-7.2) will multiply in the same fashion as the symmetry operations if T — SR, then D(T) — D(S)D(R), (see Appendix A.5-5). The P(P) so found, therefore form an n-dimensional representation of both the point group and the group of transformation operators Om, and the functions. .. and fn are said to be a basis for the representation. 

The operators just described will leave the scalar product of two functions of the function space unchanged (O,/, O,/,) = (/,-./ ). Such operators are said to be unitary and they can always be represented by unitary matrices (see 6-4). The proof that the 0M are unitary follows from considering 

Our definition of 0M applied to functions of the coordinates xx, x and x% of a point in physical space, but it can be generalized to apply to functions of any number of variables, as long as we know how those variables change under the symmetry operations. For example, if we let X stand for a complete specification of the coordinates of all the electrons (or all the nuclei) of some molecule, i.e. 

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