complex conjugate vector space について

.
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 ''J'' (different multiplication by ''i'').
==Motivation==
If

V\,

and

W\,

are complex vector spaces, a function

f\colon V \to W\,

is antilinear if
:

f(v+v') = f(v) + f(v')\quad\text\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 \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 opposite ring so that a right

R

-module can be regarded as a left

R^

-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^ \to D

.

抄文引用元・出典: フリー百科事典『ウィキペディア（Wikipedia）』
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