Complex linear analysis

What is a complex linear function?

A complex-linear functional on V. (also called a linear functional) is a (complex) linear transformation φ : V → C.
A real-linear functional on a complex (or real) vector space V is a function ψ : V → R such that ψ(x + y) = ψ(x) + ψ(y) and ψ(αx) = αψ(x) for all x, y ∈ V and all α ∈ R..

What is a complex linear transformation?

A linear transformation on a complex vector space V is a function or map that respects vector addition.
That is, given two vectors x,yu220.

1. V, a linear map f acts term by term: f(x+y)=f(x)+f(y)

1. V, a linear map f acts term by term: f(x+y)=f(x)+f(y)