  Metric algebras are metric variants of $\Sigma$-algebras. They are first
introduced in the field of universal algebra to deal with algebras equipped
with metric structures such as normed vector spaces. Recently a similar notion
of quantitative algebra is used in computer science to formulate computational
effects of probabilistic programs. In this paper we show that varieties of
metric algebras (classes defined by a set of metric equations) are exactly
classes closed under (metric versions of) subalgebras, products and quotients.
This result implies the class of normed vector spaces cannot be defined by
metric equations, differently from the classical case where the class of vector
spaces is equational. This phenomenon suggests that metric equations are not a
very natural class of formulas since they cannot express such a typical class
of metric algebras. Therefore we need a broader class of formulas to acquire an
appropriate metric counterpart of the classical variety theory.
