nLab anti-dual linear space




If the ground ring (ground field) is equipped with a star structure (an anti-involution, such as complex conjugation in the complex numbers), then the operation of forming dual linear spaces may be “twisted” by this involution to yield a notion of “anti-dual” spaces. These relate to Hermitian forms as ordinary dual spaces relate to ordinary inner products.


Let 𝕂\mathbb{K} be a star-ring, hence a unital ring (often a field, whence our notation, and here often the complex numbers) equipped with an anti-involution (for instance complex conjugation):

()¯:𝕂𝕂 \overline{(-)} \,\colon\, \mathbb{K} \to \mathbb{K}
xy¯=y¯x¯. \overline{x \cdot y} \,=\, \overline{y} \cdot \overline{x} \,.

Consider a 𝕂\mathbb{K}-module 𝒱𝕂Mod\mathscr{V} \,\in\, \mathbb{K}Mod (say right modules for possibly non-commutative 𝕂\mathbb{K}, but being just 𝕂\mathbb{K}-vector spaces if 𝕂\mathbb{K} is a field).


The anti-dual space 𝒱 *¯\overline{\mathscr{V}^\ast} of 𝒱\mathscr{V} is the space of anti-linear maps to the ground ring (ground field):

𝒱 *¯{ϕ:𝒱𝕂|v,v𝒱ϕ(v+v)=ϕ(v)+ϕ(v),v𝒱λ𝕂ϕ(vλ)=λ¯v} \overline{\mathscr{V}^\ast} \;\coloneqq\; \Big\{ \phi \,\colon\, \mathscr{V} \to \mathbb{K} \,\Big\vert\, \underset{v, v' \in \mathscr{V}}{\forall} \, \phi(v + v') = \phi(v) + \phi(v') ,\; \underset{ { v \in \mathscr{V} } \atop { \lambda \in \mathbb{K} } }{\forall} \, \phi(v \cdot \lambda) \,=\, \overline{\lambda} \cdot v \Big\}

equipped itself with the structure of a (right) module by

ϕ𝒱 *¯,λ𝕂(ϕλ)(c)=ϕ(v)λ. \phi \in \overline{\mathscr{V}^\ast} ,\, \,\lambda \in \mathbb{K} \;\;\; \vdash \;\;\; (\phi \cdot \lambda)(c) \;=\; \phi(v) \cdot \lambda \,.

(eg. Karoubi & Villamayor 1973, Ex. 1; Karoubi 2010, §1)

As the notation already indicates, an equivalent definition is:


The anti-dual 𝒱 *¯\overline{\mathscr{V}^\ast} has as underlying abelian group the ordinary linear dual 𝒱 *\mathscr{V}^\ast but equipped with its anti-linear structure, namely with the (right) module action twisted by the involution as:

ϕHom 𝕂(𝒱,𝕂)v𝒱(ϕλ)(v)=λ¯ϕ(v). \phi \,\in\, Hom_{\mathbb{K}}(\mathscr{V}, \mathbb{K}) \;\;\; \vdash \;\;\; \underset{v \in \mathscr{V}}{\forall} \, (\phi \cdot \lambda)(v) \;=\; \overline{\lambda} \cdot \phi(v) \,.

(eg. Mishchenko 1976 §1.1).

An isomorphism between the two definitions

𝒱 * 𝒱 * ϕ ϕ¯ \array{ \mathscr{V}^\ast &\longrightarrow& \mathscr{V}^\ast \\ \phi &\mapsto& \overline{\phi} }

is established by postcomposing linear forms with the star-involution

ϕ¯:vϕ(v)¯ \overline{\phi} \,\colon\, v \,\mapsto\, \overline{\phi(v)}

which respects the above (right) 𝕂\mathbb{K}-actions due to

(vϕ(v)λ) (vϕ(v)λ¯) = (vλ¯ϕ(v)¯). \array{ \big( v \,\mapsto\, \phi(v)\cdot \lambda \big) &\mapsto& \big( v \,\mapsto\, \overline{\phi(v) \cdot \lambda} \big) \\ & = & \big( v \,\mapsto\, \overline{\lambda} \cdot \overline{\phi(v)} \big) \mathrlap{\,.} }


Discussion in the context of Hermitian K-theory:

See also:

Last revised on October 25, 2023 at 08:20:08. See the history of this page for a list of all contributions to it.