Complex conjugate of a vector space

If ${\displaystyle V}$ and ${\displaystyle W}$ are complex vector spaces, a function ${\displaystyle f:V\to W}$ is antilinear if

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

With the use of the conjugate vector space ${\displaystyle {\overline {V}}}$, an antilinear map ${\displaystyle f:V\to W}$ can be regarded as an ordinary linear map of type ${\displaystyle {\overline {V}}\to W.}$ The linearity is checked by noting:

$${\displaystyle f(\alpha *v)=f({\overline {\alpha }}\cdot v)={\overline {\overline {\alpha }}}\cdot f(v)=\alpha \cdot f(v)}$$

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

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

A linear map ${\displaystyle f:V\to W\,}$ gives rise to a corresponding linear map ${\displaystyle {\overline {f}}:{\overline {V}}\to {\overline {W}}}$ which has the same action as ${\displaystyle f.}$ Note that ${\displaystyle {\overline {f}}}$ preserves scalar multiplication because

$${\displaystyle {\overline {f}}(\alpha *v)=f({\overline {\alpha }}\cdot v)={\overline {\alpha }}\cdot f(v)=\alpha *{\overline {f}}(v)}$$

Thus, complex conjugation ${\displaystyle V\mapsto {\overline {V}}}$ and ${\displaystyle f\mapsto {\overline {f}}}$ define a functor from the category of complex vector spaces to itself.

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

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

The double conjugate ${\displaystyle {\overline {\overline {V}}}}$ is identical to ${\displaystyle V.}$

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

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

• Antidual space – Conjugate homogeneous additive mapPages displaying short descriptions of redirect targets
• Linear complex structure – Mathematics concept
• Riesz representation theorem – Theorem about the dual of a Hilbert space
• conjugate bundle

• 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).