  In this paper we have found a necessary and sufficient condition for
equivalence of two norms on a linear space using the theory of exponential
vector space. Exponential vector space is an ordered algebraic structure which
can be considered as an algebraic ordered extension of vector space. This
structure is axiomatised on the basis of the intrinsic properties of the
hyperspace $\mathscr{C}(\mathcal X)$ comprising all nonempty compact subsets of
a Hausdorff topological vector space $\mathcal X$. Exponential vector space is
a conglomeration of a semigroup structure, a scalar multiplication and a
compatible partial order. We have shown that the collection of all norms
defined on a linear space, together with the constant function zero, forms a
topological exponential vector space. Then using the concept of comparing
function (a concept defined on a topological exponential vector space) we have
proved the aforesaid necessary and sufficient condition; also we have proved
using comparing function that in an infinite dimensional linear space there are
uncountably many non-equivalent norms.
