On two conditional forms of the equation∥f(x + y)∥ = ∥f(x) + f(y)∥

Summary

With reference to the functional equation

$$\left\| {f(x + y)} \right\| = \left\| {f(x) + f(y)} \right\|$$

forf: X → E, whereX andE are real linear spaces andE = (E, ∥·∥) is normed, we shall consider the following conditional equations

$$(x,y) \in X \times X,f(x + y) \ne 0 \Rightarrow \left\| {f(x + y)} \right\| = \left\| {f(x) + f(y)} \right\|$$

((1.1))

$$(x,y) \in X \times X,f(x) + f(y) \ne 0 \Rightarrow \left\| {f(x + y)} \right\| = \left\| {f(x) + f(y)} \right\|,$$

((1.2))

corresponding to the known Mikusinski and Dhombres conditional forms of the Cauchy equationf(x + y) = f(x) + f(y), respectively.

In this paper, without any regularity conditions onf, we shall prove that each of (a) and (b) is equivalent to the Cauchy equation, provided that (E, ∥·∥) is a strictly normed real space.