Math 641-600 Midterm Review — Fall 2023

The in-class part of the midterm will consist of the following: statements of theorems and definitions; short problems and propositions similar to homework problems or examples done in class; and either a critical part or sketch of a proof for one of the major theorems proved. The take-home test will have longer computations, proofs, or problems. It will be due at 4 pm on Monday, 10/16/2023.

I will also have extra office hours: Wednesday, 10/11/2023, from 11:30-12:30 and 3-4; Thursday, 10/12/2023, from 2-3:30.

Inner product spaces

• Definitions of inner product spaces and normed spaces
• Schwarz's inequality, triangle inequality
• Orthogonal projections, minimization problems, least squares, normal equations
• Gram-Schmidt process

Banach spaces and Hilbert spaces

• Convergent sequences, Cauchy sequences, complete spaces - Hilbert spaces and Banach spaces.
• Special (complete) spaces — $\ell^p, L^p\ (1 \le p \le \infty),\ C[a,b], C^k[a,b]$, and Sobolev space $ H^1[a,b]$. Be able to prove completeness for the spaces given in the notes or in homework assignments.

Lebesgue Integration

• Definitions: Lebesgue measure, Lebesgue integral, simple function, sets of measure 0
• Be able to define the Lebesgue integral using simple functions.
• Be able to state and use these theorems: Monotone convergence theorem and dominated convergence theorem

Orthonormal sets and expansions

• 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 L², orthogonal polynomials; be able to establish completeness for specific sets of orthogonal polynomials.

Approximation of continuous functions

• Modulus of continuity
• Linear spline approximation
• Weierstrass Approximation Theorem. Be able to sketch a proof, given necessary properties of the Bernstein polynomials.