Complex conjugate of a vector space

In mathematics, the complex conjugate of a complex vector space is a complex vector space that has the same elements and additive group structure as but whose scalar multiplication involves conjugation of the scalars. In other words, the scalar multiplication of satisfies where is the scalar multiplication of and is the scalar multiplication of The letter stands for a vector in is a complex number, and denotes the complex conjugate of [1]

More concretely, the complex conjugate vector space is the same underlying real vector space (same set of points, same vector addition and real scalar multiplication) with the conjugate linear complex structure (different multiplication by ).

Motivation

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If   and   are complex vector spaces, a function   is antilinear if   With the use of the conjugate vector space  , an antilinear map   can be regarded as an ordinary linear map of type   The linearity is checked by noting:   Conversely, any linear map defined on   gives rise to an antilinear map on  

This is the same underlying principle as in defining the opposite ring so that a right  -module can be regarded as a left  -module, or that of an opposite category so that a contravariant functor   can be regarded as an ordinary functor of type  

Complex conjugation functor

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A linear map   gives rise to a corresponding linear map   that has the same action as   Note that   preserves scalar multiplication because   Thus, complex conjugation   and   define a functor from the category of complex vector spaces to itself.

If   and   are finite-dimensional and the map   is described by the complex matrix   with respect to the bases   of   and   of   then the map   is described by the complex conjugate of   with respect to the bases   of   and   of  

Structure of the conjugate

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

The double conjugate   is identical to  

Complex conjugate of a Hilbert space

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Given a Hilbert space   (either finite or infinite dimensional), its complex conjugate   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 on   is an inner multiplication to some fixed vector, and vice versa.[citation needed]

Thus, the complex conjugate to a vector   particularly in finite dimension case, may be denoted as   (v-dagger, a row vector that is the conjugate transpose to a column vector  ). In quantum mechanics, the conjugate to a ket vector   is denoted as   – a bra vector (see bra–ket notation).

See also

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References

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  1. ^ K. Schmüdgen (11 November 2013). Unbounded Operator Algebras and Representation Theory. Birkhäuser. p. 16. ISBN 978-3-0348-7469-4.

Further reading

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  • 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).