Thom: Infinite-dimensional Vector Spaces

„Let f: U –> V be a map between two differential manifolds, and to each point u of U associate a variation δv of the image v=f(u). Taking all possible variations δv(u), depending differentiably on u, gives all maps g close to f, and these variations (regarded as vectors tangent to f) form an infinite-dimensional vector space. The function space L(U, V) of maps from U to V can thus be considered as an infinite-dimensional manifold. I refer to books on functional analysis for definitions and properties of the topologies (Hilbert space, Banach space, Frechet space, etc.) that are possible on infinite-dimensional vector spaces and, in turn, on infinite- dimensional manifolds. Here we are interested only in a closed subspace L of finite codimension, namely, the bifurcation set H. In the study of subspaces of this type, the actual choice of topology on the space of tangent vectors δv(u) is, in practice, irrelevant.“ (p. 339) #Thom #infinite-dimensionalVectorSpaces

Thom, René, Structural Stability and Morphogenesis: An Outline of a General Theory of Models. Boca Raton, FL: CRC Press 2018.