Agenda
Things we need to decide
What do we do in this course?
Normed Linear Spaces
Metric Spaces
Examples from geometry
Euclidean Metric
Spherical metric
Normed Linear Spaces
Continuity
Topology of a metric space
Compact subsets of a finite dimensional vector space
Continuous Linear Maps
Bounded Linear Maps
Equivalence of Norms on Finite Dimensional Vector Spaces
Completeness, Contractions, Consequences
Lipschitz Maps
The Classification Question
Series in Banach Spaces
Contraction Principle
Applications of Contraction Theorem
Exponential, Logarithm and other Functions
Bounded Linear maps as a normed linear space
The General Linear Group of a Banach Space
The exponential of a linear map
Logarithm and other functions
Differentiation
The GΓ’teaux Derivative
The FrΓ©chet Derivative
GΓ’teaux Differential as Velocity
Elementary Properties of GΓ’teaux/ FrΓ©chet Derivative
Application: Conservation of Angular Momentum
Comparison of GΓ’teaux and FrΓ©chet derivative
The Jacobian (Operator) Matrix
Differentiating Maps to a Direct Sum
Differentiating Maps defined on an Open Subset of a Direct Sum
Combining the Two: Maps from an Open Subset of a Direct Sum to a Direct Sum
The Integral
The Cauchy Integral
Characterization of Regulated Functions
Characterization of the Cauchy Integral
The Riemann Integral
Properties of the Integral
Taylor's Formula
Fundamental Theorem of Calculus
Symmetries of Higher Derivatives
Taylor's Formula
Applications I: Linear Lie Groups
Properties of the Exponential Map
Lie Algebra of a Linear Lie Group
Lie Algebras of Some Linear Lie Groups
Lie Group Lie Algebra Correspondence
BCH Formula
Integrating Lie Algebra Homomorphisms
Simply Connected Linear Lie Groups
Isomorphism Theorems
Inverse Function Theorem
Corollaries of Inverse Function Theorem
Implicit Function Theorem
Ordinary Differential Equations
Reduction to time/parameter independent case
The time dependent case
The time and parameter dependent case
Existence and Uniqueness of Flows for Lipschitz Vector Fields
Differentiability of Dependence on Initial Conditions
The Flow Equation
Vector Fields as Derivations
Applications: Optimization
Unconstrained Otimizaton
Critical Points and Gradient Flow
Constrained Optimization: Necessary Condition
Constrained Optimization: Sufficient Condition
Geometric Description of Lagrange Rule
Change of Variables in Multiple Integrals
Multilinear Algebra
tensor Products
G-Spaces
Alternating/Exterior/Wedge Product of Vector Spaces
The Wedge/Antisymmetric product Map
A Second Approach to Wedge Product Map
Pairing Between Alternating Forms and Antisymmetric Product
Appendix Axiom of Choice and All That!
Appendix Hahn-Banach Theorems
Appendix Homotopy
Fundamental Group
Covering Space
Miscellaneous Exercises