  The quotient shape types of normed vectorial spaces(over the same field) with
respect to Banach spaces reduce to those of Banach spaces. The finite quotient
shape type of normed spaces is an invariant of the (algebraic) dimension, but
not conversely. The converse holds for separable normed spaces as well as for
the bidual-like spaces (isomorphic to their second dual spaces). As a
consequence, the Hilbert space $l_2$, or even its (countably dimensional,
unitary) direct sum subspace may represent the unique quotient shape type of
all $2^\aleph_0$-dimensional normed spaces. An application yields two extension
type theorems.
