  We introduce and study holomorphically finitely generated (HFG) Fr\'echet
algebras, which are analytic counterparts of affine (i.e., finitely generated)
$\mathbb C$-algebras. Using a theorem of O. Forster, we prove that the category
of commutative HFG algebras is anti-equivalent to the category of Stein spaces
of finite embedding dimension. We also show that the class of HFG algebras is
stable under some natural constructions. This enables us to give a series of
concrete examples of HFG algebras, including Arens-Michael envelopes of affine
algebras (such as the algebras of holomorphic functions on the quantum affine
space and on the quantum torus), the algebras of holomorphic functions on the
free polydisk, on the quantum polydisk, and on the quantum polyannulus.
