Triangle Inequality for Finite Sums of Vectors

Let $V$ be a \reftext{def:vector-space-2026a}{complex vector space} equipped with a \reftext{def:complex-normed-space-2026a}{norm} $\lVert\cdot\rVert$, let $n$ be a \reftext{def:natural-numbers-2026a}{natural number}, let $[n]$ be the \reftext{def:initial-segment-natural-numbers-2026a}{initial segment} determined by $n$, and let $v:[n]\to V$ be a map with values $v_{k}$. Then

where the sum on the left is the \reftext{def:finite-sum-vector-space-2026a}{finite sum in $V$}, the sum on the right is the \reftext{def:finite-sum-field-2026b}{finite sum} in the field of \reftext{def:real-numbers-c54-2026c}{real numbers}, and the order is that of the \reftext{def:ordered-field-c54-2026b}{ordered field} of real numbers.