Editing Dual space (section)

=== Bilinear products and dual spaces ===

If ''V'' is finite-dimensional, then ''V'' is isomorphic to ''V''<sup>∗</sup>. But there is in general no [[natural isomorphism]] between these two spaces.<ref>{{harvnb|Mac Lane|Birkhoff|1999|loc=§VI.4}}</ref>  Any [[bilinear form]] {{math|{{langle}}·,·{{rangle}}}} on ''V'' gives a mapping of ''V'' into its dual space via

:<math>v\mapsto \langle v, \cdot\rangle</math>

where the right hand side is defined as the functional on ''V'' taking each {{math|''w'' ∈ ''V''}} to {{math|{{langle}}''v'', ''w''{{rangle}}}}.  In other words, the bilinear form determines a linear mapping

:<math>\Phi_{\langle\cdot,\cdot\rangle} : V\to V^*</math>

defined by

:<math>\left[\Phi_{\langle\cdot,\cdot\rangle}(v), w\right] = \langle v, w\rangle.</math>

If the bilinear form is [[nondegenerate form|nondegenerate]], then this is an isomorphism onto a subspace of ''V''<sup>∗</sup>.
If ''V'' is finite-dimensional, then this is an isomorphism onto all of ''V''<sup>∗</sup>.  Conversely, any isomorphism <math>\Phi</math> from ''V'' to a subspace of ''V''<sup>∗</sup> (resp., all of ''V''<sup>∗</sup> if ''V'' is finite dimensional) defines a unique nondegenerate bilinear form {{math|<math> \langle \cdot, \cdot \rangle_{\Phi} </math>}} on ''V'' by

:<math> \langle v, w \rangle_\Phi = (\Phi (v))(w) = [\Phi (v), w].\,</math>

Thus there is a one-to-one correspondence between isomorphisms of ''V'' to a subspace of (resp., all of) ''V''<sup>∗</sup> and nondegenerate bilinear forms on ''V''.

If the vector space ''V'' is over the [[complex numbers|complex]] field, then sometimes it is more natural to consider [[sesquilinear form]]s instead of bilinear forms.
In that case, a given sesquilinear form {{math|{{langle}}·,·{{rangle}}}} determines an isomorphism of ''V'' with the [[Complex conjugate vector space|complex conjugate]] of the dual space

: <math>
    \Phi_{\langle \cdot, \cdot \rangle} : V\to \overline{V^*}.
  </math>
The conjugate of the dual space <math>\overline{V^*}</math> can be identified with the set of all additive complex-valued functionals {{math|''f'' : ''V'' → '''C'''}} such that
: <math>
    f(\alpha v) = \overline{\alpha}f(v).
  </math>