Sequence ell-one versus nonatomic L-one for the RNP

Verify norm separability of the sequence space. Over $\mathbb{R}$, let $D$ be the finite-support sequences with rational coordinates; over $\mathbb{C}$, use coordinates in $\mathbb{Q}+i\mathbb{Q}$. For each support length the coordinate choices form a finite product of countable sets, and the union over all lengths is countable by [L1]. Given $a\in {\mathrm{\ell }}^1$ and $\epsilon >0$, first choose a finite truncation within $\epsilon \mathrm{/}2$ in ${\mathrm{\ell }}^1$, then approximate its finitely many coordinates so that the sum of coordinate errors is below $\epsilon \mathrm{/}2$. Thus $D$ is countable and dense, and ${\mathrm{\ell }}^1$ is norm separable.

Put the sequence-space side under the separable-dual theorem. By [L2], ${\mathrm{\ell }}^1$ is isometrically the continuous dual $c_0^{\ast }$ and is Banach. Step 1.1 supplies norm separability, so [L3], under the assumed AC, gives RNP to ${\mathrm{\ell }}^1$.

Contrast the nonatomic function space and audit scope. [A1, L4, step 2.1] The theorem [L4] gives the opposite conclusion for the real and complex Lebesgue quotient spaces $L^1([0,1],\lambda )$. This is not a contradiction: ${\mathrm{\ell }}^1$ consists of summable scalar sequences and is the separable dual $c_0^{\ast }$, whereas the second space is built over a nonatomic measure and has the explicit nondifferentiable indicator curve used in [L4]. The zero sequence and zero function occur in both spaces but do not determine a global geometric property. Both scalar fields are covered, and AC is propagated to [L3] and [L4], with Countable Choice used in the countability calculation of step 1.1.