In algebra, a linear topology on a left formula_1-module formula_2 is a topology on formula_2 that is invariant under translations and admits a fundamental system of neighborhood of formula_4 that consists of submodules of formula_5 If there is such a topology, formula_2 is said to be linearly topologized. If formula_1 is given a discrete topology, then formula_2 becomes a topological formula_1-module with respect to a linear topology.
The notion is used more commonly in algebra than in analysis. Indeed, "[t]opological vector spaces with linear topology form a natural class of topological vector spaces "over discrete fields", analogous to the class of locally convex topological vector spaces over the normed fields of real or complex numbers in functional analysis."
The term "linear topology" goes back to Lefschetz' work.
See also.
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