A Fréchet space is a complete, metrizable, locally
convex topological vector space. Completeness
is taken with respect to a metric 
that induces its topology and
satisfies 
for every 
, 
, and 
in the space. Its topology
is defined by a countable separating family
of seminorms.
For example, the space of smooth functions on ![[0,1]](/images/equations/FrechetSpace/Inline6.svg)
, with continuous derivatives of every order up to the endpoints, is a
Fréchet space. Its 
topology is given by the countable family of seminorms
![||f||_alpha=sup_(x in [0,1])|D^alphaf(x)|,](/images/equations/FrechetSpace/NumberedEquation1.svg) |
where 
ranges over the nonnegative integers and 
denotes the supremum.
For a Cauchy sequence 
in this topology, each derivative sequence 
converges uniformly
to a continuous function 
. Passing to the limit
in the fundamental theorems of calculus
shows that 
. Thus 
is smooth and 
converges to 
in the stated topology, so
the space is complete.
