Let "V" be a (possibly-infinite-dimensional) simple formula_1-module. If "V" happens to admit a formula_140-weight vector formula_141, then it is unique up to scaling and is called the highest weight vector of "V". It is also an formula_37-weight vector and the formula_37-weight of formula_141, a linear functional of formula_37, is called the highest weight of "V". The basic yet nontrivial facts then are (1) to each linear functional formula_146, there exists a simple formula_1-module formula_148 having formula_149 as its highest weight and (2) two simple modules having the same highest weight are equivalent. In short, there exists a bijection between formula_59 and the set of the equivalence classes of simple formula_1-modules admitting a Borel-weight vector.