Basic Analysis – Gently Done Topological Vector Spaces

10: Fréchet Spaces 113
Definition 10.2 A sequence (x n ) in a topological vector space (X,T) is said to be
a Cauchy sequence if for any neighbourhood U of 0 there is some N ∈ N such that
x n −x m ∈ U whenever n ≥ m ≥ N.
Proposition 10.3 Suppose that (X,T) is a topological vector space such that
the topology T is induced by a translation invariant metric d. Then a sequence
(x n ) is a Cauchy sequence in (X,T) if and only if it is a Cauchy sequence with
respect to the metric d in the usual metric space sense. In particular, if d and
d ′ are translation invariant metrics both inducing the same vector topology on a
vector space X, then they have the same Cauchy sequences. Furthermore, (X,d)
is complete if and only if (X,d ′ ) is complete.
Proof Suppose that (X,T) is a topologicalvector space such that T isgiven by the
translation invariant metric d. Then for any ε > 0, the ball B ε = {x : d(x,0) < ε}
is an open neighbourhood of 0, and every neighbourhood of 0 contains such a
ball. Given any sequence (x n ) in X, we have d(x n − x m ,0) = d(x n ,x m ) so that
x n − x m ∈ B ε if and only if d(x n ,x m ) < ε and the first part of the proposition
follows.
Now if d and d ′ are two invariant metrics both inducing the same topology, T,
then they clearly have the same Cauchy sequences, namely those sequences which
are Cauchy sequences in (X,T).
Suppose that (X,d) is complete, and suppose that (x n ) is a Cauchy sequence
in (X,d ′ ). Then (x n ) is a Cauchy sequence in (X,T) and hence also in (X,d).
Hence it converges in (X,d) and therefore also in (X,T), to the same limit. But
then it also converges in (X,d ′ ), i.e., (X,d ′ ) is complete.
This result means that completeness depends only on the topology determined
by the family of seminorms and not on the particular family itself. For example,
the topologywould remainunaltered if every seminorm of the familywere replaced
by some non-zero multiple of itself, say, p n were to be replaced by s n p n , where
s n > 0. The translation invariant metric d is changed by this, but the family of
Cauchy sequences (and convergent sequences) remains unchanged.
Definition 10.4 A Fréchet space is a topological vector space whose topology is
given by a countable separating family of seminorms and such that it is complete
as a metric space with respect to the translation invariant metric as defined above
via the family of seminorms.
As noted above, completeness does not depend on any particular translation
invariant metric which may induce the vector topology.
Example 10.5 A Banach space is a prime example of a Fréchet space; the vector
topology is determined by the collection of seminorms consisting of just the norm,