Infinite Dimensional Vector Space Math

In other words if its matrix representation is a column matrix with only finitely many nonzero entries.

Infinite dimensional vector space math. They re obviously infinite dimensional. Prove that k infty is infinite dimensional. Even in an infinite dimensional vector space linear combination always means finite linear combination. Such a vector space is said to be of infinite dimension or infinite dimensional.

Smith we have proven that every nitely generated vector space has a basis. I find them amazing. However i studied the vector space of all real functions und the polynomial space. I want it more specific.

We will now see an example of an infinite dimensional vector space. Does such a vector space have a basis. I ve been studying about infinite dimensional vector spaces lately. It is not difficult to check that p is indeed a vector space with the usual polynomial addition as vector addition and multiplying a polynomial by a.

Let p denote the vector space of all polynomials in x with rational coefficients. Therefore since the basis spans every element in is represented as a finite linear combination of basis elements. A finite dimensional vector space is always isomorphic to its dual but this is false for an infinite dimensional vector space. Suppose mathbb f infty is finite dimensional with dimension n let v i be the vector in mathbb f infty consisting of all 0 s except.

By de nition a basis for a vector space v. It s just infinite is boring. But what about vector spaces that are not nitely generated such as the space of all continuous real valued functions on the interval 0 1. A vector space that is not not finite dimensional is called an infinite dimensional vector space.