Complex conjugate of a 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 $\alpha \,$.

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 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 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).