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darij grinberg
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Non continuous Linear form on $E=C([0,1],\mathbb{R})$ without AC

Let's note $E=C([0,1],\mathbb{R})$ the Banach space of real continuous funtions from the [0,1] interval with the uniform norm.

Is it possible to show a non-continuous linear form on $E$ exists without using a basis, i.e. without AC?