Math 641-600 Midterm Review, Fall 2016

The midterm will consist of the following: statements of theorems and definitions; short problems or propositions similar to homework problems or examples done in class; and either a proof, or a sketch of a proof, for one of the major theorems covered in class.

Linear algebra

Inner products & norms

• Subspaces, orthogonal complements
• Orthogonal sets of vectors, the Gram-Schmidt procedure
• Least squares, minimization problems, projections, normal equations

Self-adjoint matrices & their properties

• Spectral theorem
• Estimation of eigenvalues
  • Maximum principle
  • The Courant-Fischer theorem
• The Fredholm Alternative

Function spaces

Banach spaces and Hilbert spaces

• Convergent sequence, Cauchy sequence, complete spaces - Hilbert spaces and Banach spaces
• Special (complete) spaces - $\ell^p$, $L^p$, $C[a,b]$, $C^k[a,b]$, Sobolev space $H^1[a,b]$

Lebesgue Integration

• Lebesgue integral, sets of measure 0 and L^p spaces
• Density of continuous functions in L^p[a,b], 1 ≤ p < ∞
• Monotone convergence theorem and dominated convergence theorem

Orthonormal sets and expansions

• Minimization problems, least squares, normal equations
• Complete sets of orthogonal/orthonormal functions, Parseval's identity, other conditions equivalent to completeness of a set
• Dense sets and completeness
• Completeness of polynomials in $L^2$, orthogonal polynomials

Approximation of continuous functions

• Modulus of continuity, linear spline approximation
• Bernstein polynomials
• Weierstrass Approximation Theorem