The direct product ∏ ∏ i ∈ ∈ I V i of a family of vector spaces V i consists of the set of all tuples ( v i ) i ∈ ∈ I , which specify for each index i in some index set I an element v i of V i . Addition and scalar multiplication is performed componentwise. A variant of this construction is the direct sum ⨁ ⨁ i ∈ ∈ I V i (also called coproduct and denoted ∐ ∐ i ∈ ∈ I V i ), where only tuples with finitely many nonzero vectors are allowed. If the index set I is finite, the two constructions agree, but in general they are different.