The dual vector space 
of a vector space 
over a field 
is the vector space of linear transformations 
, with pointwise addition
and scalar multiplication.
For a real vector space, these linear functionals take values in 
. For a complex vector space,
they take values in the complex numbers.
If 
is finite-dimensional, its dual vector space has the
same dimension as 
. Given a vector basis 
, ..., 
for 
there exists a dual basis for
, written 
, ..., 
, where 
and 
is the kronecker delta.
For finite-dimensional 
, another way to realize an isomorphism
with 
is through an inner product. A real
vector space can have a symmetric inner
product 
in which case a vector 
corresponds to a dual element
by 
.
Then a vector basis corresponds to its dual
basis only if it is an orthonormal basis,
in which case 
.
A complex vector space can have a Hermitian
inner product, in which case 
is a conjugate-linear isomorphism
of 
with 
,
i.e., 
.
Dual vector spaces can describe many objects in linear algebra. When 
and 
are finite-dimensional vector
spaces, an element of the vector
space tensor product 
, say 
, corresponds to the linear
transformation 
.
That is, 
.
For example, the identity map is 
. A bilinear
form on 
,
such as a real inner product, is an element
of 
.
A complex Hermitian inner product is a
sesquilinear form rather than complex bilinear.
