Return to Question

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 linear form on $E$ without using a basis, i.e. 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?