Banach Manifolds

Abstract

Typical examples of manifolds are sufficiently smooth curves and surfaces in ℝⁿ which have a tangent space (tangent line, tangent plane) at each point. Manifolds will always be manifolds without boundary. One may think, for example, of the surface of a ball. Manifolds with boundary, such as the ball itself, will be considered in Section 73.19.

The categories of differentiable manifolds and vector bundles provide a useful context for the mathematics needed in mechanics, especially the new topological and qualitative results.

Ralph Abraham and Jerrold Marsden (1978)

Too often in the physical sciences, the space of states is postulated to be a linear space when the basic problem is essentially nonlinear; this confuses the mathematical development.

Steve Smale (1980)

The proof that … is left as a masochistic exercise for the reader. Rest assured that we will never have to do this sort of abstract nonsense.

Michael Spivak (1979)