Equivalently

Definition 1. X is a linear vector space, if one has the operations (x, y) → x+y and (λ, x) → λx. We say that l: X → R is linear functional if l(λx+µy) = λl(x) +µl(y). Definition 2. Let L be a linear space. The linear span of a set S ⊂ L, is a linear subspace of L defined by span[S] = { N∑ j=1 λjxj, λj ∈ F, xj ∈ S}