Complex conjugate vector space

In mathematics, the (formal) complex conjugate of a complex vector space $V\,$ is the complex vector space $\overline V$ consisting of all formal complex conjugates of elements of $V\,$. That is, $\overline V$ is a vector space whose elements are in one-to-one correspondence with the elements of $V\,$:

$\overline V = \{\overline v \mid v \in V\},$

with the following rules for addition and scalar multiplication:

$\overline v + \overline w = \overline{\,v+w\,}\quad\text{and}\quad\alpha\,\overline v = \overline{\,\overline \alpha \,v\,}.$

Here $v\,$ and $w\,$ are vectors in $V\,$, $\alpha\,$ is a complex number, and $\overline\alpha$ denotes the complex conjugate of $\alpha\,$.

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

In the case where $V\,$ is a linear subspace of $\mathbb{C}^n$, the formal complex conjugate $\overline V$ is naturally isomorphic to the actual complex conjugate subspace of $V\,$ in $\mathbb{C}^n$.

Antilinear maps

If $V\,$ and $W\,$ are complex vector spaces, a function $f\colon V \to W\,$ is antilinear if

$f(v+v') = f(v) + f(v')\quad\text{and}\quad f(\alpha v) = \overline\alpha \, f(v)$

for all $v,v'\in V\,$ and $\alpha\in\mathbb{C}$.

One reason to consider the vector space $\overline V$ is that it makes antilinear maps into linear maps. Specifically, if $f\colon V \to W\,$ is an antilinear map, then the corresponding map $\overline V \to W$ defined by

$\overline v \mapsto f(v)$

is linear. Conversely, any linear map defined on $\overline V$ gives rise to an antilinear map on $V\,$.

One way of thinking about this correspondence is that the map $C\colon V \to \overline V$ defined by

$C(v) = \overline v$

is an antilinear bijection. Thus if $f\colon \overline V \to W$ if linear, then composition $f \circ C\colon V \to W\,$ is antilinear, and vice versa.

Conjugate linear maps

Any linear map $f \colon V \to W\,$ induces a conjugate linear map $\overline f \colon \overline V \to \overline W$, defined by the formula

$\overline f (\overline v) = \overline{\,f(v)\,}.$

The conjugate linear map $\overline f$ is linear. Moreover, the identity map on $V\,$ induces the identity map $\overline V$, and

$\overline f \circ \overline g = \overline{\,f \circ g\,}$

for any two linear maps $f\,$ and $g\,$. Therefore, the rules $V\mapsto \overline V$ 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 $\mathcal B$ of $V\,$ and $\mathcal C$ of $W\,$, then the map $\overline f$ is described by the complex conjugate of $A\,$ with respect to the bases $\overline{\mathcal B}$ of $\overline V$ and $\overline{\mathcal C}$ of $\overline W$.

Structure of the conjugate

The vector spaces $V\,$ and $\overline V$ 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 $\overline V$. (The map $C\,$ is not an isomorphism, since it is antilinear.)

The double conjugate $\overline{\overline V}$ is naturally isomorphic to $V\,$, with the isomorphism $\overline{\overline V} \to V$ defined by

$\overline{\overline v} \mapsto v.$

Usually the double conjugate of $V\,$ is simply identified with $V\,$.

Complex conjugate of a Hilbert space

Given a Hilbert space $\mathcal{H}$ (either finite or infinite dimensional), its complex conjugate $\overline{\mathcal{H}}$ is the same vector space as its continuous dual space $\mathcal{H}'$. There is one-to-one antilinear correspondence between continuous linear functionals and vectors. In other words, any continuous linear functional on $\mathcal{H}$ 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 ^* (v-star, 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

• Linear complex structure

References

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