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 vector space 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).

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 $f\colon \overline V \to W$ is 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).

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