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Comparison of numerical analysis software


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 this 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, KL and KR. In that case, the most general linear combination looks like{\displaystyle a_{1}v_{1}b_{1}+\cdots +a_{n}v_{n}b_{n}\,}

a_{1}v_{1}b_{1}+\cdots +a_{n}v_{n}b_{n}\,

where a1,…,an belong to KLb1,…,bn belong to KR, and v1,…,vn belong to V.