Complex conjugate of a vector space

In mathematics, the complex conjugate of a complex vector space $$V\,$$ is a complex vector space $$\overline V$$ that 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 $$\alpha\,*\, v = {\,\overline{\alpha} \cdot \,v\,}$$ where $$*$$ is the scalar multiplication of $$\overline{V}$$ and $$\cdot$$ is the scalar multiplication of $$V.$$ The letter $$v$$ stands for a vector 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 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 $$f(v + w) = f(v) + f(w) \quad \text{ and } \quad f(\alpha v) = \overline{\alpha} \, f(v)$$ 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 $$\overline{V} \to W.$$ The linearity is checked by noting: $$f(\alpha * v) = f(\overline{\alpha} \cdot v) = \overline{\overline{\alpha}} \cdot f(v) = \alpha \cdot f(v)$$ Conversely, any linear map defined on $$\overline{V}$$ gives rise to an antilinear map on $$V.$$

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

Complex conjugation functor
A linear map $$f : V \to W\,$$ gives rise to a corresponding linear map $$\overline{f} : \overline{V} \to \overline{W}$$ that has the same action as $$f.$$ Note that $$\overline f$$ preserves scalar multiplication because $$\overline{f}(\alpha * v) = f(\overline{\alpha} \cdot v) = \overline{\alpha} \cdot f(v) = \alpha * \overline{f}(v)$$ Thus, complex conjugation $$V \mapsto \overline{V}$$ 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 double conjugate $$\overline{\overline{V}}$$ is identical to $$V.$$

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 $$\mathcal{H}^{\prime}.$$ 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 $$v^\dagger$$ (v-dagger, a row vector that 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).