Vector Spaces Without Fields
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High Quality Content by WIKIPEDIA articles! In mathematics, the conventional definition of the concept of vector space relies upon the algebraic concept of a field. This article treats an algebraic definition that does not require that concept. If vector spaces are redefined as (universal) algebras as below, no preliminary introduction of fields is necessary. On the contrary, fields will come from such vector space algebras. One of the ways to do this is to keep the first four Abelian group axioms on addition in the standard formal definition and to formalize its geometric idea of scaling only...
High Quality Content by WIKIPEDIA articles! In mathematics, the conventional definition of the concept of vector space relies upon the algebraic concept of a field. This article treats an algebraic definition that does not require that concept. If vector spaces are redefined as (universal) algebras as below, no preliminary introduction of fields is necessary. On the contrary, fields will come from such vector space algebras. One of the ways to do this is to keep the first four Abelian group axioms on addition in the standard formal definition and to formalize its geometric idea of scaling only by notions that concern every universal algebra. Vector space algebras consist of one binary operation + and of "unary" operations , which form a nonempty set , that satisfy the following conditions, which do not involve fields.
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