theory of semisimple Lie algebras, the Weyl character formula is an additional result established after the representations have been classified. In Weyl's analysis of the compact group case, however, the Weyl character formula is actually a crucial part of the classification itself. Specifically, in Weyl's analysis of the representations of K, the hardest part of the theorem—showing that every dominant, analytically integral element is actually the highest weight of some representation—is proved in a totally different way from the usual Lie algebra construction using Verma modules. In Weyl's approach, the construction is based on the Peter–Weyl theorem and an analytic proof of the Weyl character formula. Ultimately, the irreducible representations of K are realized inside the space of continuous functions on K. The SU(2) case We now consider the case of the compact group SU(2). The representations are often considered from the Lie algebra point of view, but we here look at them from the group point of view. We take the maximal torus to be the set of matrices of the form According to the example discussed above in the section on representations of T, the analytically integral elements are labeled by integers, so that the dominant, analytically integral elements are non-negative integers . The general theory then tells us that for each , there is a unique irreducible representation of SU(2) with highest weight . Much information about the representation corresponding to a given is encoded in its character. Now, the Weyl character formula says, in this case, that the character is given by We can also write the character as sum of exponentials as follows: (If we use the formula for the sum of a finite geometric series on the above expression and simplify, we obtain the earlier expression.) From this last expression and the standard formula for the character in terms of the weights of the representation, we can read off that the weights of the representation are each with multiplicity one. (The weights are the integers appearing in the exponents of the exponentials and the multiplicities are the coefficients of the exponentials.) Since there are weights, each with multiplicity 1, the dimension of the representation is . Thus, we recover much of the information about the representations that is usually obtained from the Lie algebra computation. An outline of the proof We now outline the proof of the theorem of the highest weight, following the