Existence of Highest Weight Vectors for Compact Connected Lie Groups identifies how weights organize finite-dimensional representations of compact Lie groups. In concrete terms, it formalizes the statement that let be a compact connected Lie group with gT G t g C:= g R C t C:= t R CU(1):=\z C:|z|=1\X^(T):=(T,U(1)) t C^R t C^ g C t C g\0\R^+ RV1, and let be a smooth finite-dimensional comp.