Linear Combination - Generalizations

Generalizations

If V is a topological vector space, then there may be a way to make sense of certain infinite linear combinations, using the topology of V. For example, we might be able to speak of a₁v₁ + a₂v₂ + a₃v₃ + ..., going on forever. Such infinite linear combinations do not always make sense; we call them convergent when they do. Allowing more linear combinations in this case can also lead to a different concept of span, linear independence, and basis. The articles on the various flavours of topological vector spaces go into more detail about these.

If K is a commutative ring instead of a field, then everything that has been said above about linear combinations generalizes to this case without change. The only difference is that we call spaces like V modules instead of vector spaces. If K is a noncommutative ring, then the concept still generalizes, with one caveat: Since modules over noncommutative rings come in left and right versions, our linear combinations may also come in either of these versions, whatever is appropriate for the given module. This is simply a matter of doing scalar multiplication on the correct side.

A more complicated twist comes when V is a bimodule over two rings, K_L and K_R. In that case, the most general linear combination looks like

where a₁,...,a_n belong to K_L, b₁,...,b_n belong to K_R, and v₁,...,v_n belong to V.