Discuss Infinite Dimensional And Finite Dimensional Vector Spaces.

Discuss Infinite Dimensional And Finite Dimensional Vector Spaces.

Finite Dimensional: A vector space is finite-dimensional if it has a basis with only finitely many vectors. (One reason for sticking to finite-dimensional spaces is so that the representation of a vector with respect to a basis is a finitely-tall vector, and so can be easily written.) We shall take the term "vector space" to mean "finite-dimensional vector space". Other spaces are interesting and important, but they lie outside of our scope.

Infinite Dimensional: The process of extending the algebraic and geometrical methods of linear algebra from matrices to differential or integral operators consists of going from a finite dimensional vector space, typically Rⁿ , to an infinite dimensional vector space, typically a function space. However, a vector space of functions has certain idiosyncrasies precisely because its dimension is infinite. These peculiarities are so important that we must develop the framework in which they arise.