6 1. BASIC NOTIONS
Example 1.10. Let [[t]] be the formal power series ring and U, V (discrete)
-vector spaces. Let us show that each [[t]]-linear map f : [[t]] U [[t]] V
is automatically continuous.
We verify this fact by proving that, for each sequence {an}1

converging to
a [[t]] U, the sequence f(an)

1
converges to f(a) in [[t]] V . The con-
vergence {an}0

a means that, for each n 0 there exists k 1 such that
a ak
(tn)
U. It is obvious from the description of [[t]] U in terms of power
series with coefficients in U given in Example 1.7 that the last condition in fact
says that a ak is divisible by
tn:
a ak =
tn
· uk
n,
for some uk
n
[[t]] U.
We conclude that f(a) f(ak)
(t)n
V , which shows that f(an)

1
converges
to f(a) in the topology of [[t]] V .
We leave as an exercise based on Theorem 11.22 of [AM69] to prove the fol-
lowing generalization of Example 1.10.
Proposition 1.11. Let R be a
regular3
local complete Noetherian ring and U,
V discrete vector spaces. Then each R-linear map f : R U R V is continuous.
Topologized tensor products. Suppose we are given a ring S and topological
S-modules M and N. One may topologize the tensor product M ⊗S N by requiring
that the subspaces
(1.8) M ⊗S V + U ⊗S N M ⊗S N,
where U (resp. V) are open S-submodules of M (resp. N), form a basis of open
neighborhoods of zero in M ⊗S N. This topology has a certain universal property
with respect to uniformly continuous maps which we formulate below. Let us recall
some necessary definitions.
A uniformity on a set X is a system of neighborhoods of the diagonal
Δ(X) := (x, x) | x X X × X
satisfying suitable axioms [Kel55, Chapter 6]. A set with a uniformity is called
a uniform space. Each uniformity induces a topology on X, with a basis of open
neighborhoods of x X given by the sets
Ux := {x X | (x , x) U},
where U .
A map f : (X, ) (Y, ) is uniformly continuous if, for each V , there
exists U such that
(x1,x2) U =⇒
(
f(x1),f(x2)
)
V.
Each uniformly continuous map is continuous with respect to the induced topologies.
The cartesian product X1
× X2 of uniform spaces (Xi,
i
), i = 1, 2, has a uni-
formity
1
×
2
given by the subsets
U1 × U2 (X1 × X1) × (X2 × X2)

=
(X1 × X2) × (X1 × X2) = Δ(X1 × X2),
where Ui i. The induced topology on X1 × X2 is the product of the induced
topologies.
3See
[AM69, Theorem 11.22] for a definition of regular local rings.
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