MAT 544 Real Analysis

3. The Implicit Function Theorem and its special case, the Inverse Function Theorem. Relation to the Contraction Mapping Fixed Point Theorem. Computation of several iterates of the approximating sequence to the implicit function. Continuity and differentiability of the implicit function. Application to differentiable variation of the fixed point in the Contraction Mapping Fixed Point Theorem. 4. The Picard-Lindelöf Theorem on existence and uniqueness of solutions of a first-order ODE whose defining equation is locally Lipschitz. Relation to the Contraction Mapping Fixed Point Theorem. Maximal extensions. 5. Diagonalization of diagonalizable matrices; Jordan form for matrices of small size. The characteristic polynomial. Eigenvalues. Eigenvectors, eigenspaces and generalized eigenspaces. The semisimple-nilpotent decomposition of a matrix. Algorithm for 2×2, 3×3 and 4×4 matrices. 6. Matrix exponentials. Definition via convergent power series. Compatibility with M 7→ UMU−1; compatibility for commuting matrices. Computation given the Jordan form. 7. Solutions of constant coefficient linear systems of ODEs, both homogeneous and inhomogeneous. Reduction of order. Solution of first-order systems by matrix exponentials. “Green’s functions” / “fundamental solutions”; solution of the inhomogeneous equation. 8. Algebras and σ-algebras in the power set of a set. The Borel σ-algebra. Product σ-algebras. 9. Measures, outer measures and premeasures. Finite and countable additivity. Subadditivity. Monotonicity. Continuity from above and below. Caratheordory’s Theorem constructing a measure from an outer measure. Compatibility with the premeasure used to construct an outer measure.