1. BASIC NOTIONS 7

Important examples of uniform spaces are provided by topological linear spaces.

Such a space X is uniform, with the uniformity given by the sets

(x1,x2) ∈ X × X | x1 − x2 ∈ U ,

where U runs over a basis of open neighborhoods of 0 in X. It is clear that the

topology induced by is the original one. A linear map between linear topological

spaces is continuous if and only if it is uniformly continuous.

Let us finally formulate the universal property of the topology (1.8).

Proposition 1.12. Suppose that M, N and Z are topological S-modules. The

map F : M ⊗S N → Z induced by an S-bilinear map F : M × N → Z is continuous

if and only if F is uniformly continuous.4

Proof. The continuity of F means by definition that, for each open neigh-

borhood W of 0 in Z, there exist open S-linear subspaces U ⊂ M and V ⊂ N

such that

(1.9) F(M ⊗S V + U ⊗S N) ⊂ W.

By the definition of F and its linearity one has

F (M ⊗S V + U ⊗S N) = F(M ⊗S V) + F (U ⊗S N)

= SpanS

(

F (M, V)

)

+ SpanS

(

F (U,N)

)

,

where SpanS(−) denotes the S-linear envelope. We see that (1.9) is equivalent to

(1.10) F (M, V) ⊂ W & F (U,N) ⊂ W.

On the other hand, the uniform continuity of F means that, for each open

neighborhood W of 0 in Z, there exist neighborhoods U ⊂ M and V ⊂ N such that

(1.11) x − x ∈ U & y − y ∈ V =⇒ F (x , y ) − F (x , y ) ∈ W.

One has, by bilinearity,

F (x , y ) − F (x , y ) = F (x − x , y ) + F (x , y − y ).

Assume the inclusions (1.10). Then F (x − x , y ) ∈ W because F (U,N) ⊂ W, and

F (x , y − y ) ∈ W because F (M, V) ⊂ W. So (1.10) implies (1.11).

Putting y = y = y in (1.11) we get that

F (x , y) − F (x , y) = F (x − x , y) ∈ W,

whenever x − x ∈ U, therefore F (U,N) ⊂ W. The same argument establishes the

inclusion F (M, V) ⊂ W, thus (1.11) implies (1.10). The lemma is proved.

The tensor product − ⊗S − with the topology (1.8) is suited for studying mul-

tilinear maps and adequate to our purposes. It makes the category of topological

S-modules a symmetric monoidal category with the unit object S. This, by defini-

tion, means the existence of natural isomorphisms of topological S-modules

S ⊗S M

∼

=

M ⊗S S

∼

=

M (unitality),

M ⊗S N

∼

= N ⊗S M (commutativity),

M ⊗S (N ⊗S O)

∼

= (M ⊗S N) ⊗S O (associativity),

4Bilinear

maps are not linear, so their continuity does not imply their uniform continuity.