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Linear Space


There are at least two distinct notions of linear space throughout mathematics.

The term linear space is most commonly used within functional analysis as a synonym of the term vector space.

The term is also used to describe a fundamental notion in the field of incidence geometry. In particular, a linear space is a space S=(p,L) consisting of a collection p={p_alpha} of points and a set L={lambda_alpha} of lines subject to the following axioms:

1. Any two distinct points of S belong to exactly one line of S.

2. Any line of S has at least two points of S.

3. There are at least three points of S not on a common line.

Using this terminology, lines are considered to be "distinguished subsets" of the collection p of points. Moreover, in this context, one can view a linear space as a generalization of the notions of projective space and affine space (Batten and Beutelspracher 2009).


See also

Affine Space, Incidence Axioms, Projective Space, Vector Space

This entry contributed by Christopher Stover

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References

Batten, L. M. and Beutelspracher, A. The Theory of Finite Linear Spaces: Combinatorics of Points and Lines. New York: Cambridge University Press, 2009.Royden, H. L. and Fitzpatrick, P. M. Real Analysis. Pearson, 2010.

Cite this as:

Stover, Christopher. "Linear Space." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/LinearSpace.html

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