# Continuous linear functionals and the Axiom of Choice

Can one prove without the Axiom of Choice that for every normed vector space $$X$$ there exist a nonzero continuous linear functional on $$X$$?

The answer is no. Let $$\ell^\infty$$ be the space of bounded sequences equipped with the $$\sup$$ norm, and $$c_0$$ its closed subspace of sequences convergent to zero. The quotient $$\ell^\infty/c_0$$ is then a Banach space when equipped with the quotient norm, and it is consistent that it does not have any nonzero continuous functionals - see a discussion in this answer.
• See also Martin Väth's paper "The dual space of $L_\infty$ is $L_1$", Indagationes Math. 9, No. 4, 619--625 (1998); Zbl 0922.46066. Oct 16 at 9:39