Editing Quotient space (linear algebra) (section)

== Quotient of a Banach space by a subspace ==
If ''X'' is a [[Banach space]] and ''M'' is a [[closed set|closed]] subspace of ''X'', then the quotient ''X''/''M'' is again a Banach space. The quotient space is already endowed with a vector space structure by the construction of the previous section. We define a [[norm (mathematics)|norm]] on ''X''/''M'' by
:<math> \| [x] \|_{X/M} = \inf_{m \in M} \|x-m\|_X = \inf_{m \in M} \|x+m\|_X = \inf_{y\in [x]}\|y\|_X. </math>

=== Examples ===
Let ''C''[0,1] denote the Banach space of [[continuous function|continuous]] real-valued [[function (mathematics)|functions]] on the [[interval (mathematics)|interval]] [0,1] with the [[sup norm]]. Denote the subspace of all functions ''f'' &isin; ''C''[0,1] with ''f''(0) = 0 by ''M''. Then the equivalence class of some function ''g'' is determined by its value at 0, and the quotient space {{nowrap|''C''[0,1]/''M''}} is isomorphic to '''R'''.

If ''X'' is a [[Hilbert space]], then the quotient space ''X''/''M'' is isomorphic to the [[Hilbert space#Orthogonal complements and projections|orthogonal complement]] of ''M''.

=== 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>