In mathematics, the complex conjugate of a complex vector space but whose scalar multiplication involves conjugation denotes the complex conjugate

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).

Motivation

is antilinear can be regarded as an ordinary linear map

This is the same underlying principle as in defining the opposite ring-module-module, or that of an opposite category so that a contravariant functor

Complex conjugation functor

define a functor from the category of complex vector spaces to itself.

is described by the complex matrix with respect to the bases

Structure of the conjugate

have the same dimension over the complex numbers and are therefore isomorphic as complex vector spaces. However, there is no natural isomorphism

Complex conjugate of a Hilbert space

Given a Hilbert space is the same vector space as its continuous dual space There is one-to-one antilinear correspondence between continuous linear functionals and vectors. In other words, any continuous linear functional is an inner multiplication to some fixed vector, and vice versa.

(v-dagger, a row vector that is the conjugate transpose). In quantum mechanics, the conjugate to a ketvector – a bra vector (see bra–ket notation).

See also

• conjugate bundle

References

Further reading

• Budinich, P. and Trautman, A. The Spinorial Chessboard. Springer-Verlag, 1988. . (complex conjugate vector spaces are discussed in section 3.3, pag. 26).