Outline and homework

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MATH 1270, Fall 2008
Outline and homework assignments

Date Sections What is covered. Comments Homework assignments

9/3 What is Functional Analysis is about. Metric spaces.

9/5 1.1, 1.2 Completion of metric spaces.
Vector spaces, normed spaces, Banach spaces
Examples: $C[a,b], l^p$

9/8 s 1.2, Mikowski's, Holder and Young inequalities

9/10 s 1.5 Resonance Lemma (cf s. 3.2). Completeness of $\ell^p$ 1. Prove that $C[a,b]$ is complete.
2. Prove that $C_p[a,b]$ is not complete
3. s 1.6, # 12, 13

9/12 s. 4.1 Bounded linear operators

9/15 $\epsilon/3$ -theorem and the uniform boundedness principle.   See notes

9/17 s. 2.1 Hilbert spaces. Basic definitions and examples s. 2.4 # 1--4.

9/19 s. 2.1 Orthogonal bases.

9/22 s. 2.2 Orthogonal projections. Orthogonal decomposition Click here to see the assignment. (will be colected Fri 9/26)

9/24 see notes Fourier series. Definition, motivation. Dirichlet kernel. Show that $\|D_n\|_{L^1} \sim \log n$ . (see notes for details). As a warm-up, estimate $\sum_{n=1}^{10^{23}} \frac{1}{n}$ .
This assignment will be discussed Fri. 9/26..

9/26 See notes Convergence of Fourier series; convergence in $L^2$ . Divergence at a point. Fejer means.

9/29 See notes Uniform convergence of Fejer means.

10/1 See notes Convergence of Fejer means in $[; L^p ;]$ . Properties of convolution, convolution theorem. Dini condition, Riemann--Lebesgue lemma and pointwise convergence of Fourier series.

10/3 s. 2.3 Riesz Representation theorem. s 2.4 # 11, 14 d, e (we are considering $[; L^2(-\pi, \pi) ;]$ here).

10/6 see handout and notes Adjoint operator. Self-adjoint, normal and unitary operators.

10/8 Click here to get the assignment.

10/10

10/13 Columbus day, no classes

10/15

10/17

10/20

10/22

10/24

10/27

10/29

10/31

11/3

11/5

11/7

11/10

11/12

11/14

11/17

11/19

11/21

11/24

11/26

11/28 Thanksgiving recess

12/1

12/3

12/5

12/8

12/10

Date	Sections	What is covered. Comments	Homework assignments

9/3		What is Functional Analysis is about. Metric spaces.
9/5	1.1, 1.2	Completion of metric spaces. Vector spaces, normed spaces, Banach spaces Examples: $C[a,b], l^p$

9/8	s 1.2,	Mikowski's, Holder and Young inequalities
9/10	s 1.5	Resonance Lemma (cf s. 3.2). Completeness of $\ell^p$	1. Prove that $C[a,b]$ is complete. 2. Prove that $C_p[a,b]$ is not complete 3. s 1.6, # 12, 13
9/12	s. 4.1	Bounded linear operators

9/15		$\epsilon/3$ -theorem and the uniform boundedness principle.	See notes
9/17	s. 2.1	Hilbert spaces. Basic definitions and examples	s. 2.4 # 1--4.
9/19	s. 2.1	Orthogonal bases.

9/22	s. 2.2	Orthogonal projections. Orthogonal decomposition	Click here to see the assignment. (will be colected Fri 9/26)
9/24	see notes	Fourier series. Definition, motivation. Dirichlet kernel.	Show that $\\|D_n\\|_{L^1} \sim \log n$ . (see notes for details). As a warm-up, estimate $\sum_{n=1}^{10^{23}} \frac{1}{n}$ . This assignment will be discussed Fri. 9/26..
9/26	See notes	Convergence of Fourier series; convergence in $L^2$ . Divergence at a point. Fejer means.

9/29	See notes	Uniform convergence of Fejer means.
10/1	See notes	Convergence of Fejer means in $[; L^p ;]$ . Properties of convolution, convolution theorem. Dini condition, Riemann--Lebesgue lemma and pointwise convergence of Fourier series.
10/3	s. 2.3	Riesz Representation theorem.	s 2.4 # 11, 14 d, e (we are considering $[; L^2(-\pi, \pi) ;]$ here).

10/6	see handout and notes	Adjoint operator. Self-adjoint, normal and unitary operators.
10/8			Click here to get the assignment.
10/10

10/13		Columbus day, no classes
10/15
10/17

10/20
10/22
10/24

10/27
10/29
10/31

11/3
11/5
11/7

11/10
11/12
11/14

11/17
11/19
11/21

11/24
11/26
11/28		Thanksgiving recess

12/1
12/3
12/5

12/8
12/10

MATH 1270, Fall 2008 Outline and homework assignments

MATH 1270, Fall 2008
Outline and homework assignments