16.-Linear-Algebra-by-Kenneth-Hoffman · Problem 2.1.6

Vector Spaces

Mathematics — Linear Algebra · vector-space-of-functions

Problem

Let

V

be the set of all complex-valued functions

f

on the real line such that (for all

t

R

)

f(-t) = \overline{f(t)}.

The bar denotes complex conjugation. Show that

V

, with the operations

(f + g)(t) = f(t) + g(t)

(cf)(t) = cf(t)

is a vector space over the field of real numbers. Give an example of a function in

V

which is not real-valued.