---
tag: 46MSX1FXZT
kind: concept
name: Convenient vector space
msc: [46A17, 58C20, 46T20]
specializes: [B6GY2KJMWW]
sources:
  - url: https://en.wikipedia.org/wiki/Convenient_vector_space
same_as:
  wikidata: Q25303670
  wikipedia: Convenient vector space
---

## Summary

A convenient vector space is a [[locally convex space|B6GY2KJMWW]] that is complete in a weak sense: every Mackey–Cauchy sequence converges.
These spaces are the setting of the convenient calculus of Frölicher, Kriegl and Michor.
In it, a map is smooth when it sends smooth curves to smooth curves.
The category of convenient spaces and smooth maps is cartesian closed.

This makes spaces of smooth maps, and manifolds of mappings, easy to handle without Banach-space estimates.

## Explanations

### L1 human

Calculus beyond Banach spaces has a problem: there are many inequivalent notions of a derivative.
The convenient approach tests smoothness only along curves: a map is smooth if every smooth path is sent to a smooth path.
For this to work, the space must contain enough limits, at least those that integrating a smooth curve produces.
Convenient vector spaces are the spaces with exactly this much completeness.
Every Banach space and every [[Fréchet space|3GAA614673]] is convenient.