Given a vector space V, it is often possible to form another vector space by taking a subset S of V and using the operations of V. Since V is a vector space, the operations of addition and scalar multiplication always produce another vector in V. For a new system using a subset S of V as its universal set to be a vector space, the set S must be closed under the operations of addition and scalar multiplication. That is, the sum of two elements of S must always be an element of S, and the product of a scalar and an element of S must always be an element of S.
S is a subset of . If
is any element of S and is any scalar, then
is also an element of S. If