of : , where all , and multiplying a linear combination of by a scalar will produce another linear combination of : . Thus is a subspace of .

:It follows that , since every is a linear combination of (trivially). Suppose that is a linear subspace of containing . Since is closed under addition and scalar multiplication, then every linear combination must be contained in . Thus, is contained in every subspace of containing , and the intersection of all such subspaces, or the smallest such subspace, is equal to the set of all