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

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

a) Show that

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

is a linear subspace of $K^I$.

b) For every $i\in I$, let $e_i\in K^I$ be defined by

$e_i(j)=\{\begin{array}{c}1,\text{ if }j=i,\\ 0\text{ else}.\end{array}$

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

Create a solution