The quotient vector space of a vector space 
by a subspace 
consists of vectors
modulo 
.
For example, suppose that 
and 
. Then the quotient space 
(read as "
mod 
") is isomorphic to 
.
In general, when 
is a subspace of a vector
space 
,
the quotient space 
is the set of equivalence
classes ![[v]](/images/equations/QuotientVectorSpace/Inline13.svg)
where 
if 
.
By "
is equivalent to 
modulo 
,"
it is meant that 
for some 
in 
,
and is another way to say 
. In particular, the elements
of 
represent ![[0]](/images/equations/QuotientVectorSpace/Inline24.svg)
.
Sometimes the equivalence classes ![[v]](/images/equations/QuotientVectorSpace/Inline25.svg)
are written as cosets 
.
The quotient space is an abstract vector space. By extending a vector basis of 
to one of 
, one obtains a complementary subspace isomorphic to 
, but this complement need not be canonical.
If 
is finite-dimensional with an inner
product, then 
is isomorphic to
 |
In the example above, with the standard inner product, 
.
Unfortunately, a different choice of inner product can change 
.
In the infinite-dimensional case, if 
is a Hilbert space, it is
necessary and sufficient
for 
to be a closed subspace
to identify each coset uniquely with its orthogonal representative
in 
,
as well as to ensure that 
with its quotient topology
is a T2-space.
