INFINITE- DIMENSIONAL SPACE

Here are two common but unhelpful was to think about infinity.
1. Infinity makes things harder.
2. Infinity is a useless academic abstraction.
Neither of these is necessarily true. Problems are often formulated in terms of infinity to make things easier and to solve realistic problems. Infinity is usually a simplification. Think of infinity as “so big I don’t have to worry about how big it is.”
Infinite dimensional spaces guide our thinking about large finite dimensional spaces. If you want to solve a practical problem in high dimensions, the infinite dimensional case may be a better guide ordinary three dimensional space. Continuity in infinite dimensional spaces requires structure that may not be apparent in low dimensions. Thinking about the infinite case may prepare you to exploit that structure in a large finite dimensional problem.
In mathematics and physics, the dimension of a space or object is informally defined as the minimum number of coordinates needed to specify each point within it. Thus a line has a dimension of one because only one coordinate is needed to specify a point on it. A surface such as a plane or the surface of a cylinder or sphere has a dimension of two because two coordinates are needed to specify a point on it (for example, to locate a point on the surface of a sphere you need both its latitude and its longitude). The inside of a cube, a cylinder or a sphere is three-dimensional because three co-ordinates are needed to locate a point within these spaces.
The concept of infinite-dimensional spaces can be formulated in a meaningful way in Hilbert spaces, which are an essential concept in quantum mechanics.The mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space. It extends the methods of vector algebra and calculus from the two-dimensional Euclidean plane and three-dimensional space to spaces with any finite or infinite number of dimensions.Infinite dimensional spaces guide our thinking about large finite dimensional spaces.