Vector space

A vector space (or linear space) is the basic object of study in the branch of mathematics called linear algebra.

Generalizations and additional structures

It is common to study vector spaces with certain additional structures. This is often necessary for recovering ordinary notions from geometry. Some of these additional structures include:

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• A real or complex vector space with a defined length concept, i.e., a norm, is called a normed vector space.
• A real or complex vector space with a notion of both length and angle is called an inner product space.
• A vector space with a topology compatible with the operations (i.e., such that addition and scalar multiplication are continuous maps) is called a topological vector space.
• A vector space with a bilinear operator (defining a multiplication of vectors) is an algebra over a field.

The definition of a vector space makes perfectly good sense if one replaces the field of scalars F by a general ring R. The resulting structure is called a module over R. In other words, a vector space is nothing but a module over a field.

Related Topics:
Ring - Module

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