| Topics covered | |
|---|---|
21/09 | 1. REVISION OF
VECTOR SPACES Vector space operations, bases, linear maps and associated matrices. Theorem that any finite-dim vector space is isomorphic to Rn. Spaces of infinite real sequences; definition of l1 and loo. Spaces of functions defined on a finite interval [a,b]; the sum of two functions. Examples of continuous and integrable functions defined on [a,b]. Exercises: uniqueness of basis coefficients, spaces of polynomials, the harmonic series. |
23/09 | Subspaces,
examples, evaluation of functions. The kernel of a linear
map. 2. DEFINITION OF A NORMED VECTOR SPACE Motivation: interaction between spaces of functions and convergence. Metric spaces, examples of distances, spheres and balls. Definition of a norm. |
28/09 | Lectures suspended 1030->1430 |
30/09 | Proof of the
Cauchy-Schwarz and triangle inequality for Rn. Definition
of the p-norms for Rn and C[a,b]. 3. LIMITS AND CONVERGENCE IN NORMED VECTOR SPACES Convergence of sequences, continuity of functions. Limit points and closure of a subset of a NVS. |
05/10 | Definition of
open sets. Open/closed complements. Characterization of continuity
using inverse images of open sets. 4. INEQUALITIES AND CONVEXITY Proofs of Holder and Minkowski inequalities. Definition of the p and oo norm on Rn. Convexity of the unit ball in a NVS. Exercises: Open sets and limits points: the subset {ein: n=1,2,..} of C, fact that f-1(a,b) is open if f continuous. |
07/10 | 5. CAUCHY
SEQUENCES AND COMPLETENESS Cauchy condition for a sequence. Completeness of R2, assuming that of R. The Banach spaces lp with p in [1,oo], and C[a,b] (proof deferred). The sequence tanh(nx) and incompleteness of the L2 norm. |
12/10 | Summary of normed
vector spaces and notions of convergence. Detailed investigation of
the sequence (ein); extracting a convergent
subsequence. Absolute convergence implies convergence in a Banach
space, and proof of a converse statement. Exercises: subsequeces of Cauchy sequences and limit points. 6. INNER PRODUCT SPACES Definition of a real (and complex) inner product, starting from Rn. Verification that an inner product gives rise to a norm. |
14/10 | Orthonormal sets, best approximations in finite dimensions. Bessel's inequality. The trigonometric sequence in C[0,2π]. Definition of Hilbert space. |
19/10 | 7. CONVERGENCE IN
HILBERT SPACE Convergence of an orthonormal series. Definition of a basis. Equivalence relations: basic theory then examples of (i) functions with integrals equal, (ii) Cauchy sequences whose difference converges to 0. Definition of L2[a,b] as completion of C[a,b] with the L2 norm (idea only). Exercises from Sheet 1: subspaces, convergence, estimates for ln(n). |
21/10 | 8. FURTHER
RESULTS, THE Lp SPACES. Definition of null sets, simple examples, the Cantor set. Equivalence of functions that differ on a null set, relation with the integral. Definition of the NVS Lp(A) for an interval A (ignoring measurability). Examples of (non-)integrable functions. |
26/10 | Statement of
Holder's inequality for integrals. The inclusion Lp(a,b) in
L1(a,b). Statements of density and completeness theorems
concerning C[a,b] and Lp(a,b), statement of completeness of
the trigonometric basis for Lp(-π,π), and
applications to Fourier series. 9. SUBSPACES OF HILBERT SPACE Examples of (non-)closed subspaces. Definition of Uperp. Existence of a point v0 of least norm in a closed convex subset K. Exercises on (non-)pointwise convergence of Fourier series, and Σn-2. Properties of Uperp. |
28/10 | Inner products
with v0. Proof of the Projection Theorem. 10. LINEAR FUNCTIONALS Equivalence of continuity and boundedness. Proving the set of continuous linear functionals is a vector space. |
02/11 | QUIZ on
convergence of functions. Definition of the norm of a linear functional, proof that V* is always complete, dual spaces in finite dimensions. Exercises: comments on sheet 2, examples of linear functionals on C[0,1]. |
03/11 | 11. LINEAR
MAPPINGS BETWEEN NVS's Linear maps and isometries between NVS's, boundedness, definition of the kernel. Proof that (lp)* = lq in detail. Table of other dual vector spaces. |
04/11 | The isometry
l2 -> L2[0,2π] and its relevance to Fourier
series. 12. LINEAR TRANSFORMATIONS OF A BANACH SPACE. The Banach space L(V,V). The geometric series and invertibility of I-A. Eigenvalues. The inequality λ <= ||T||. Exercise: A self-adjoint integral operator T on L2(-π.π) and its spectrum. |