Vector Space - Vector Spaces With Additional Structure

Vector Spaces With Additional Structure

From the point of view of linear algebra, vector spaces are completely understood insofar as any vector space is characterized, up to isomorphism, by its dimension. However, vector spaces per se do not offer a framework to deal with the question—crucial to analysis—whether a sequence of functions converges to another function. Likewise, linear algebra is not adapted to deal with infinite series, since the addition operation allows only finitely many terms to be added. Therefore, the needs of functional analysis require considering additional structures. Much the same way the axiomatic treatment of vector spaces reveals their essential algebraic features, studying vector spaces with additional data abstractly turns out to be advantageous, too.

A first example of an additional datum is an order ≤, a token by which vectors can be compared. For example, n-dimensional real space Rn can be ordered by comparing its vectors componentwise. Ordered vector spaces, for example Riesz spaces, are fundamental to Lebesgue integration, which relies on the ability to express a function as a difference of two positive functions

ƒ = ƒ+ − ƒ−,

where ƒ+ denotes the positive part of ƒ and ƒ− the negative part.