Complex Conjugate Vector Space

In mathematics, the complex conjugate of a complex vector space $V\,$ is a complex vector space $\overline V$, which has the same elements and additive group structure as $V,$ but whose scalar multiplication involves conjugation of the scalars. In other words, the scalar multiplication of $\overline V$ satisfies
$\alpha\,*\, v =$
where $*$ is the scalar multiplication of $\overline$ and $\cdot$ is the scalar multiplication of $V.$
The letter $v$ stands for a vector in $V,$ $\alpha$ is a complex number, and $\overline$ denotes the complex conjugate of $\alpha.$

More concretely, the complex conjugate vector space is the same underlying vector space (same set of points, same vector addition and real scalar multiplication) with the conjugate linear complex structure $J$ (different multiplication by $i$).

Motivation

If $V$ and $W$ are complex vector spaces, a function $f : V \to W$ is antilinear if
$f(v + w) = f(v) + f(w) \quad \text \quad f(\alpha v) = \overline \, f(v)$
With the use of the conjugate vector space $\overline V$, an antilinear map $f : V \to W$ can be regarded as an ordinary linear map of type $\overline \to W.$ The linearity is checked by noting:
$f(\alpha * v) = f(\overline \cdot v) = \overline \cdot f(v) = \alpha \cdot f(v)$
Conversely, any linear map defined on $\overline$ gives rise to an antilinear map on $V.$

This is the same underlying principle as in defining opposite ring so that a right $R$-module can be regarded as a left $R^$-module, or that of an opposite category so that a contravariant functor $C \to D$ can be regarded as an ordinary functor of type $C^ \to D.$

Complex conjugation functor

A linear map $f : V \to W\,$ gives rise to a corresponding linear map $\overline : \overline \to \overline$ which has the same action as $f.$ Note that $\overline f$ preserves scalar multiplication because
$\overline(\alpha * v) = f(\overline \cdot v) = \overline \cdot f(v) = \alpha * \overline(v)$
Thus, complex conjugation $V \mapsto \overline$ and $f \mapsto\overline f$ define a functor from the category of complex vector spaces to itself.

If $V$ and $W$ are finite-dimensional and the map $f$ is described by the complex matrix $A$ with respect to the bases $\mathcal$ of $V$ and $\mathcal$ of $W,$ then the map $\overline$ is described by the complex conjugate of $A$ with respect to the bases $\overline$ of $\overline$ and $\overline$ of $\overline.$

Structure of the conjugate

The vector spaces $V$ and $\overline$ have the same dimension over the complex numbers and are therefore isomorphic as complex vector spaces. However, there is no natural isomorphism from $V$ to $\overline.$

The double conjugate $\overline$ is identical to $V.$

Complex conjugate of a Hilbert space

Given a Hilbert space $\mathcal$ (either finite or infinite dimensional), its complex conjugate $\overline$ is the same vector space as its continuous dual space $\mathcal^.$
There is one-to-one antilinear correspondence between continuous linear functionals and vectors.
In other words, any continuous linear functional on $\mathcal$ is an inner multiplication to some fixed vector, and vice versa.

Thus, the complex conjugate to a vector $v,$ particularly in finite dimension case, may be denoted as $v^\dagger$ (v-dagger, a row vector which is the conjugate transpose to a column vector $v$).
In quantum mechanics, the conjugate to a ''ket vector'' $\,, \psi\rangle$ is denoted as $\langle\psi, \,$ – a ''bra vector'' (see bra–ket notation).

See also

*
*
*
* conjugate bundle

References

Further reading

* Budinich, P. and Trautman, A. ''The Spinorial Chessboard''. Springer-Verlag, 1988. {{ISBN, 0-387-19078-3. (complex conjugate vector spaces are discussed in section 3.3, pag. 26).

Linear algebra
Vector space