Editing Quotient space (linear algebra) (section)

=== Generalization to locally convex spaces ===
The quotient of a [[locally convex space]] by a closed subspace is again locally convex.<ref>{{Harvard citation text|Dieudonné|1976}} p. 65, § 12.14.8</ref>  Indeed, suppose that ''X'' is locally convex so that the [[topological space|topology]] on ''X'' is generated by a family of [[seminorm]]s {''p''<sub>α</sub>&nbsp;|&nbsp;α&nbsp;&isin;&nbsp;''A''} where ''A'' is an index set.  Let ''M'' be a closed subspace, and define seminorms ''q''<sub>α</sub> on ''X''/''M'' by

:<math>q_\alpha([x]) = \inf_{v\in [x]} p_\alpha(v).</math>

Then ''X''/''M'' is a locally convex space, and the topology on it is the [[quotient topology]].

If, furthermore, ''X'' is [[metrizable]], then so is ''X''/''M''.  If ''X'' is a [[Fréchet space]], then so is ''X''/''M''.<ref>{{Harvard citation text|Dieudonné|1976}} p. 54, § 12.11.3</ref>