Mathematical operators spacing considerations

If W is a finite matrix, linear algebra tells us that this can in fact be asserted when W is symmetric. If W is an operator in an infinite-dimensional space the mathematical conditions are considerably more complicated, but as a rule of thumb one may also regard any symmetrical operator as diagonaliz-able - as is customary in quantum mechanics. It will now be shown that the detailed balance property guarantees that the operator W is symmetrical. We adopt the notation of a continuous range. 

A third and more complex way is to first find another set of operators 0M, which have certain fundamental properties and are homomorphic with the symmetry operations, and then find a set of matrices which are homomorphic with these new operators. This last step is achieved by consideration of the effect that the 0M have on some family of mathematical functions (a so-called function space) e.g. a set of five d-orbitals it will be seen that the choice of the function space and the choice of the 0M are bound up with each other. It is important to realize that this method involves two steps as opposed to the first two methods which involve only one. 

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