Product set/Arbitrary/Vector spaces/Direct sum/Definition

Direct sum

Let ${}I$ denote a set, and let ${}K$ denote a field. Suppose that, for every ${}i\in I$, a ${}K$-vector space ${}V_{i}$ is given. Then the set

${}\bigoplus _{i\in I}V_{i}={\left\{(v_{i})_{i\in I}\mid v_{i}\in V_{i},\,v_{i}\neq 0{\text{ for only finitely many }}i\right\}}\,$

is called the direct sum of the ${}V_{i}$.