Editing Differential (mathematics) (section)

==== Differentials as linear maps on a vector space ====

The same procedure works on a vector space with a enough additional structure to reasonably talk about continuity. The most concrete case is a Hilbert space, also known as a [[Complete metric space|complete]] [[inner product space]], where the inner product and its associated [[Norm (mathematics)|norm]] define a suitable concept of distance. The same procedure works for a Banach space, also known as a complete [[Normed vector space]]. However, for a more general topological vector space, some of the details are more abstract because there is no concept of distance.

For the important case of a finite dimension, any inner product space is a Hilbert space, any normed vector space is a Banach space and any topological vector space is complete. As a result, you can define a coordinate system from an arbitrary basis and use the same technique as for <math>\mathbb{R}^n</math>.