Question

Give an example of a function φ : C −→ C such that φ(w + z) = φ(w) + φ(z) For all w, z ∈ C but not linear. (Here C is thought of as complex vector space.)

Answer 1

An example of a function φ : C → C that satisfies φ(w + z) = φ(w) + φ(z) for all w, z ∈ C but is not linear is the complex conjugation function.

φ(w) = w*

Where w* is the complex conjugate of w. This function preserves addition but is not linear because it does not satisfy φ(cw) = cφ(w) for all complex numbers c.

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