Let $K$ be a \reftext{def:field-c54-2026b}{field}, let $n$ be a \reftext{def:natural-numbers-2026a}{natural number}, let $a_{k}$ and $b_{k}$ be elements of $K$ for each natural number $k$ with $1\le k\le n$, and let $\lambda\in K$. All sums below are the \reftext{def:finite-sum-field-2026a}{finite sums} of that definition. Then the following hold.

\textbf{1. (Additivity)}

\textbf{2. (Homogeneity)}

\textbf{3. (Conjugation)} If $K$ is the field of \reftext{def:complex-numbers-2026a}{complex numbers}, then, with the \reftext{def:complex-conjugate-2026a}{complex conjugate},

\textbf{4. (Nonnegative summands)} If $K$ is the field of \reftext{def:real-numbers-c54-2026c}{real numbers} and $0\le a_{k}$ for every $k$ with $1\le k\le n$, then $0\le\sum_{k=1}^{n}a_{k}$; and if in addition $\sum_{k=1}^{n}a_{k}=0$, then $a_{k}=0$ for every $k$ with $1\le k\le n$.