Basic Analysis – Gently Done Topological Vector Spaces

10. Fréchet Spaces
Inthischapter wewillconsider completelocallyconvex topologicalvectorspaces—
this class includes, in particular, Banach spaces.
Theorem 10.1 Let (X,T) be a topological vector space with topology determined
by a countable separating family of seminorms. Then (X,T) is metrizable.
Moreover, the metric d can be chosen to be translation invariant in the sense that
d(x+a,y +a) = d(x,y) for any x,y and a in X.
Proof Let {p n : n ∈ N} be a countable separating family of seminorms which
determine the topology T on X. For any x,y in X, put
d(x,y) =
∞�
n=1
1
2 n
p n (x−y)
1+p n (x−y) .
Then d is well-defined and clearly satisfies d(x,x) = 0, d(x,y) = d(y,x) ≥ 0 and
d(x+a,y+a) = d(x,y), x,y,a ∈ X. Now, p n (x−z) ≤ p n (x−y)+p n (y−z) and
the map t ↦→ t/(1+t) is increasing on [0,∞) and so
pn (x−z)
1+p n (x−z) ≤ pn (x−y)
1+p n (x−y) + pn (y −z)
1+p n (y −z)
which implies that d(x,z) ≤ d(x,y) + d(y,z), x,y,z ∈ X. Finally, we see that
d(x,y) = 0 implies that p n (x − y) = 0 for all n, and so x = y since {p n } is a
separating family. Thus d is a translation invariant metric on X.
We must show that d induces the vector topology on X determined by the
family {p n }. To see this, we note that a net (x ν ) in X converges to x with respect
to the vector topology T if and only if (p n (x ν −x)) converges to 0 for each n ∈ N.
This is equivalent to d(x ν ,x) → 0, that is, x ν → x in the metric topology on X
induced by d. It follows that the two topologies have the same closed sets and
therefore the same open sets, that is, they coincide.
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