Supplement for Manifolds and Differential Geometry Jeffrey M. Lee

28 0. Background Material
0.7. Differential Calculus on Banach Spaces
Modern differential geometry is based on the theory of differentiable manifoldsa
natural extension of multivariable calculus. Multivariable calculus is said
to be done on (or in) an n-dimensional coordinate space R n (also called
variously “Euclidean space” or sometimes “Cartesian space”. We hope that
the great majority of readers will be comfortable with standard multivariable
calculus. A reader who felt the need for a review could do no better
than to study the classic book “Calculus on Manifolds” by Michael Spivak.
This book does multivariable calculus 3 in a way suitable for modern differential
geometry. It also has the virtue of being short. On the other hand,
calculus easily generalizes from R n to Banach spaces (a nice class of infinite
dimensional vector spaces). We will recall a few definitions and facts
from functional analysis and then review highlights from differential calculus
while simultaneously generalizing to Banach spaces.
A topological vector space over R is a vector space V with a topology
such that vector addition and scalar multiplication are continuous. This
means that the map from V × V to V given by (v 1 , v 2 ) ↦→ v 1 + v 2 and the
map from R × V to V given by (a, v) ↦→ av are continuous maps. Here we
have given V × V and R × V the product topologies.
Definition 0.34. A map between topological vector spaces which is both
a continuous linear map and which has a continuous linear inverse is called
a toplinear isomorphism.
A toplinear isomorphism is then just a linear isomorphism which is also
a homeomorphism.
We will be interested in topological vector spaces which get their topology
from a norm function:
Definition 0.35. A norm on a real vector space V is a map ‖.‖ : V → R
such that the following hold true:
i) ‖v‖ ≥ 0 for all v ∈ V and ‖v‖ = 0 only if v = 0.
ii) ‖av‖ = |a| ‖v‖ for all a ∈ R and all v ∈ V.
iii) If v 1 , v 2 ∈ V, then ‖v 1 + v 2 ‖ ≤ ‖v 1 ‖+‖v 2 ‖ (triangle inequality). A vector
space together with a norm is called a normed vector space.
Definition 0.36. Let E and F be normed spaces. A linear map A : E −→ F
is said to be bounded if
‖A(v)‖ ≤ C ‖v‖
for all v ∈ E. For convenience, we have used the same notation for the norms
in both spaces. If ‖A(v)‖ = ‖v‖ for all v ∈ E we call A an isometry. If
3 Despite the title, most of Spivak’s book is about calculus rather than manifolds.