Editing Bilinear map (section)

=== Composition map ===
{{See also|Topology of uniform convergence}}

Let <math>X, Y, \text{ and } Z</math> be [[locally convex]] [[Hausdorff space]]s and let <math>C : L(X; Y) \times L(Y; Z) \to L(X; Z)</math> be the composition map defined by <math>C(u, v) := v \circ u.</math> 
In general, the bilinear map <math>C</math> is not continuous (no matter what topologies the spaces of linear maps are given). 
We do, however, have the following results:

Give all three spaces of linear maps one of the following topologies:
# give all three the topology of bounded convergence; 
# give all three the [[topology of compact convergence]]; 
# give all three the [[topology of pointwise convergence]].

* If <math>E</math> is an [[equicontinuous]] subset of <math>L(Y; Z)</math> then the restriction <math>C\big\vert_{L(X; Y) \times E} : L(X; Y) \times E \to L(X; Z)</math> is continuous for all three topologies.{{sfn | Trèves | 2006 | pp=424-426}}
* If <math>Y</math> is a [[barreled space]] then for every sequence <math>\left(u_i\right)_{i=1}^{\infty}</math> converging to <math>u</math> in <math>L(X; Y)</math> and every sequence <math>\left(v_i\right)_{i=1}^{\infty}</math> converging to <math>v</math> in <math>L(Y; Z),</math> the sequence <math>\left(v_i \circ u_i\right)_{i=1}^{\infty}</math> converges to <math>v \circ u</math> in <math>L(Y; Z).</math>{{sfn| Trèves | 2006 | pp=424-426}}