Differential calculus | Calculus of variations | Multivariable calculus

Convenient vector space

In mathematics, convenient vector spaces are locally convex vector spaces satisfying a very mild completeness condition. Traditional differential calculus is effective in the analysis of finite-dimensional vector spaces and for Banach spaces. Beyond Banach spaces, difficulties begin to arise; in particular, composition of continuous linear mappings stop being jointly continuous at the level of Banach spaces, for any compatible topology on the spaces of continuous linear mappings. Mappings between convenient vector spaces are smooth or if they map smooth curves to smooth curves. This leads to a Cartesian closed category of smooth mappings between -open subsets of convenient vector spaces (see property 6 below). The corresponding calculus of smooth mappings is called convenient calculus.It is weaker than any other reasonable notion of differentiability, it is easy to apply, but there are smooth mappings which are not continuous (see Note 1).This type of calculus alone is not useful in solving equations. (Wikipedia).

Convenient vector space
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Tangent bundle | Lie group | Vector space | Normed vector space | Multivariable calculus | Banach space | Minkowski functional | Absolutely convex set | A priori estimate | Uniform space | Exponential map (Riemannian geometry) | Differentiable manifold | Continuous linear operator | Dual space | Riemannian manifold | Polar set | Bornological space | Nash–Moser theorem | Compact space | Locally convex topological vector space | Cartesian closed category