16.-Linear-Algebra-by-Kenneth-Hoffman · Problem 8.1.14

Inner Product Spaces

Mathematics — Linear Algebra · conjugation

Problem

Let

V

be a complex vector space and

W

a subset of

V

with the following properties: (a)

W

is a real vector space with respect to the operations defined in

V

. (b) For each

\alpha

V

there exist unique vectors

\beta

\gamma

W

such that

\alpha = \beta + i\gamma

. Show that the equation

J\alpha = \beta - i\gamma

defines a conjugation on

V

such that

J\alpha = \alpha

if and only if

\alpha

belongs to

W

, and show also that

J

is the only conjugation on

V

with this property.