An infinite dimensional vector space is a mathematical concept that describes a collection of objects (vectors) that can be added and multiplied together. Unlike finite dimensional vector spaces, which have a finite number of basis vectors, infinite dimensional vector spaces have an infinite number of basis vectors.
Some properties of an infinite dimensional vector space include the ability to add and multiply vectors, as well as scalar multiplication and distributivity. Additionally, infinite dimensional vector spaces have a zero vector and additive inverses for each vector.
The main difference between an infinite dimensional vector space and a finite dimensional vector space is the number of basis vectors. In a finite dimensional vector space, there is a finite number of basis vectors, while in an infinite dimensional vector space, there is an infinite number of basis vectors.
Some examples of infinite dimensional vector spaces include function spaces, such as the space of all continuous functions on a closed interval, and sequence spaces, such as the space of all real sequences.
Infinite dimensional vector spaces are important in science because they provide a way to model and solve problems that involve an infinite number of variables or dimensions. They are also used in many areas of mathematics, such as functional analysis and differential equations, which have numerous applications in science.