The complex coordinate sum norm, including the zero space

If $n=0$, [F3], [F5] and [F6] say that the only vector is zero and its empty sum is $0$. If $n>0$ and $z=0$, pointwise equality in [F2] makes each coordinate zero; [F4] makes each summand zero, and the finite sum in [F5] is then zero. Conversely, if $N_1(z)=0$, the nonnegative summands from [F4] satisfy the zero-sum condition in [F8], so every $\mathrm{\mid }{z}_j{\mathrm{\mid }}_{\mathbb{C}}=0$ and hence every $z_j=0$ by [F4]. Pointwise equality [F1] identifies $z$ with the zero function [F2]. Thus $N_1(z)=0$ iff $z=0$ for every $n$, as required in [F9].

For every $\lambda \in \mathbb{C}$, the finite-list sum in [F5], pointwise scalar multiplication from [F2], and multiplicativity from [F4] give $N_1(\lambda z)={\sum }_{j<n}\mathrm{\mid }\lambda z_j{\mathrm{\mid }}_{\mathbb{C}}={\sum }_{j<n}\mathrm{\mid }\lambda {\mathrm{\mid }}_{\mathbb{C}}\mathrm{\mid }{z}_j{\mathrm{\mid }}_{\mathbb{C}}=\mathrm{\mid }\lambda {\mathrm{\mid }}_{\mathbb{C}}{N}_1(z)$, where the last equality is finite-sum scaling by the real scalar $\mathrm{\mid }\lambda {\mathrm{\mid }}_{\mathbb{C}}$ from [F7]. This is the complex absolute homogeneity clause in [F9].

Pointwise addition from [F2], the finite-list sum from [F5], and complex modulus subadditivity from [F4] give $\mathrm{\mid }{z}_j+w_j{\mathrm{\mid }}_{\mathbb{C}}\le \mathrm{\mid }{z}_j{\mathrm{\mid }}_{\mathbb{C}}+\mathrm{\mid }{w}_j{\mathrm{\mid }}_{\mathbb{C}}$ for every $j<n$. Finite-sum monotonicity and additivity [F7, F8] therefore give $N_1(z+w)={\sum }_{j<n}\mathrm{\mid }{z}_j+w_j{\mathrm{\mid }}_{\mathbb{C}}\le {\sum }_{j<n}(\mathrm{\mid }{z}_j{\mathrm{\mid }}_{\mathbb{C}}+\mathrm{\mid }{w}_j{\mathrm{\mid }}_{\mathbb{C}})=N_1(z)+N_1(w)$. Together with steps 1.1 and 1.2, this is the triangle clause in [F9], so all three complex norm criteria hold.