Vector space/I to K/Almost all zero/Exercise

Let ${}K$ be a field, let ${}I$ denote an index set, and let ${}K^{I}=\operatorname {Map} \,{\left(I,K\right)}$ be the corresponding vector space.

1. Show that

   ${}E={\left\{f\in K^{I}\mid f(i)=0{\text{ for all }}i\in I{\text{ up to finite exceptions}}\right\}}\,$

   is a linear subspace of ${}K^{I}$.

2. For every ${}i\in I$, let ${}e_{i}\in K^{I}$ be defined by

   ${}e_{i}(j)={\begin{cases}1,{\text{ if }}j=i\,,\\0{\text{ else}}\,.\end{cases}}\,$

   Show that every element ${}f\in E$ can be expressed uniquely as a linear combination of the family ${}e_{i}$, ${}i\in I$.

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