We construct a bounded closed convex set in c₀(C) on which no nonzero continuous complex-linear functional attains its maximum modulus. Consequently, this answers in the negative the two longstanding questions raised by Deville, Godefroy and Zizler in [p. 35]{DGZ}. The construction uses a small perturbation of the identity on c₀(R).
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Xin Luo (2026) studied this question.
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