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Solutions of some partial differential equations are obtained as criticalpoints of a real funtional. Then the Banach space where this functional isdefined has to be real, otherwise, it is not differentiable. It follows thatthe equation is solved with respect to the real dual space of this Banachspace. But if the solution is complex-valued there is the following problem:what does the multiplication of this equation by a complex number mean ? Inthis note, we explain how to rigorously define this operation.

The dual space of a complex Banach space restricted to the field of real numbers