Complex conjugate of a vector space explained

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$$\backslash alpha\backslash ,*\backslash ,\; v\; =$$where * is the scalar multiplication of \overline{V} and ⋅ 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. ^[1] J (different multiplication by i ).

Motivation

If

V and W are complex vector spaces, a function f : V \to W is antilinear if$$f(v\; +\; w)\; =\; f(v)\; +\; f(w)\; \backslash quad\; \backslash text\; \backslash quad\; f(\backslash alpha\; v)\; =\; \backslash overline\; \backslash ,\; f(v)$$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:$$f(\backslash alpha\; *\; v)\; =\; f(\backslash overline\; \backslash cdot\; v)\; =\; \backslash overline\; \backslash cdot\; f(v)\; =\; \backslash alpha\; \backslash cdot\; f(v)$$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$$\backslash overline(\backslash alpha\; *\; v)\; =\; f(\backslash overline\; \backslash cdot\; v)\; =\; \backslash overline\; \backslash cdot\; f(v)\; =\; \backslash alpha\; *\; \backslash overline(v)$$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 l{B} of V and l{C} of W, then the map \overline{f} is described by the complex conjugate of A with respect to the bases \overline{ l{B}} of \overline{V} and \overline{ l{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

l{H} (either finite or infinite dimensional), its complex conjugate \overline{ l{H}} is the same vector space as its continuous dual space l{H} ^{\prime} . There is one-to-one antilinear correspondence between continuous linear functionals and vectors. In other words, any continuous linear functional on l{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).

See also

• • • • conjugate bundle

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).

Notes and References

1. Book: K. Schmüdgen. Unbounded Operator Algebras and Representation Theory. 11 November 2013. Birkhäuser. 978-3-0348-7469-4. 16.