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**Facts
about Hilbert and Banach spaces**

**Class diary:
****As we will not follow the same order as the book, here
is an outline of the material covered in each class. **

**1.
Introduction to normed and pre-Hilbert spaces****:**

**August 25: **Introduction.
Definitions of norm and inner product. Examples of normed and
pre-Hilbert spaces. **Section I.1 **(1.1,
1.2, 1.3, 1.5, 1.7. Example 1.8 has not been seen in class, read it
in the book). **Section III.1**
(1.1, 1.2, 1.8, 1.9. Read also
Examples 1.10 and 1.11).

**August 27:** Inequalities:
Cauchy-Schwarz, Hölder's
inequality and Minkowsky's ineq for Lp spaces. Characterization of
distances arising from a norm. Interplay between the topological and
vector structure. **Section
I.1 **(1.4),
**Section
III.1** (1.3)

**August
30: **Interplay
between the topological and vector structure (continued): Continuity
of operations, closed subspaces, convex subsets, completeness.
Characterization of norms arising from an inner product. **Section
I.2 **(2.3,2.4)

**2. Hilbert
spaces:**

**September 1: **Orthogonality
and projections. **Rest of Section
I.2.**

**September 3: **Problems
about projections

**September 8: **The
Riesz Representation Theorem. **All
of Section I.3**

**September 10:**
Lax-Milgram
Theorem and problems about Riesz Representation (not in book).

**September 13-17: **Orthonormal
sets and bases. Gram-Schmidt. Bessel's Inequality. Characterization
of a basis (the Main Theorem) **Section
I.4. **

**September 20: **The
Fourier basis in L^2[0,1]. Every Hilbert space has a basis.

**September 22: **The Haar
and Walsh bases in L^2[0,1]. Cardinality of bases. Every Hilbert
space is isometrically isomorphic to a space of sequences.

**September 24: **Exercises
for Section I.4.

**3. Banach spaces:**

**September 27: **Linear
operators on Banach spaces (**Section
III.2 **(Proposition
2.1, Exercise 1). Def of Banach Algebra.

**September 29: **Equivalence
of norms (**Section III.1**
(1.5), **Section
III.3** (3.1)). Norms in
quotient spaces (**Section III.4**).

**October 1: **Norms
in quotient spaces (continued). **Hahn-Banach** and
its corollaries (**Section III.6**).
The two versions of the Hahn-Banach theorem are Thm 6.2 and Coroll.
6.4 in the book. Our Corollary 1 is 6.5 in the book, our Corollary 3
is 6.7, our Corollary 4 is 6.8. Our density criterion is Corollary
6.14.

**October 4:**
The
dual of l^1 is l^∞.
The dual of l^p is l^(p') (The book only mentions it in III.5.9 and
5.10)

**October 6:
**The
application J:X->X**. The dual of c0 is l^1. Reflexive spaces. c0,
l^1, l^∞ are
not reflexive. **(Section
III.11)**

**October 8:
**The
dual of L^p(Ω,μ)
is L^(p')(Ω,μ),1≤p**<**∞.
**(Appendix B in book)**

**October 11:**
Banach
limits **(Section
III.7)**

**October 13: **Exercises
about the bidual space.

**4. The Baire
Category Theorem and its consequences**

**October 15: **Baire's
Theorem (not in book) and some corollaries.

**October 18: **Exercises
on Baire's Theorem.

**October 20: **The
Open Mapping Theorem **(Section
III.12)**. Application to
Fourier series in L^1[0,1]

**October 22: **Fourier
series in L^1[0,1]. Definition of closed operators.

**October 25: **Two
examples of closed operators that are not continuous. The Closed
Graph Theorem. Separable spaces are isomorphic to a quotient space of
l^1.

**October 27: **Complemented
subspaces **(III.13)****. **The
Principle of Uniform Boundedness **(Section
III.14)**

**October 29: **The
Principle of Uniform Boundedness. Application to Fourier series in
L^p[0,1].

**November 1: **Fourier
series (continued)

**5.
Topological vector spaces and Weak Topologies**

**November 3: **Introduction
to weak topologies and topological vector spaces (TVS)

**November 5: **Locally
convex spaces (LCS): Seminorms and their properties.**(Section
IV.1)**

**November 8: **Examples
of LCS. Which LCS are metrizable? **(Sections
IV.1,IV.2)**

**November 10: **Metrizable
LCS continued. Lp(0,1), p<1, is a metric space without a bounded
convex open set .**(Section IV.2)**

**November 12: **Separation
properties of TVS. Normable LCS.