Week 4
HILBERT SPACES
Week 4
Lecture 4.1
INTRODUCTION
What we will learn this week:
What is an inner product?
What is a Hilbert space?
The Riesz Representation Theorem
Who is the dual space of a Hilbert space?
How to define a linear operator with a b

Week 6
LP SPACES
Week 6
Lecture 6.1
INTRODUCTION
Four weeks ago
For p [1,[, we defined Lp() the set of
measurable functions from to R
whose p-th power belongs to L1().
We defined L() the set of measurable functions
from to R for which there exists a real

Week 5
LAX MILGRAM & COMPLEMENTS
ON HILBERT SPACES
Week 5
Lecture 5.1
INTRODUCTION
What we will learn this week:
The Lax-Milgram Lemma
Strong and weak convergence in a Hilbert
Hilbertian Basis
Week 5
Lecture 5.2
CONTINUITY AND COERCIVITY OF
A BILINEAR

Week 3
BANACH SPACES &
LINEAR CONTINUOUS
FUNCTIONS
Week 3
Lecture 3.1
INTRODUCTION
What we will learn this week:
What is a Banach space?
The Fischer-Riesz theorem
What is L(X,Y)?
What is a dual space?
What is the weak topology?
Strong vs. weak converge

Week 5: Lax Milgram & Complements
on Hilbert spaces
Document prepared by Anna Rozanova-Pierrat1
1
Lecture 5.2: Continuity and coercivity of a bilinear form
We refer to Lectures 4.2, 4.3 and 4.6 for the definition of a bilinear form and its properties on a

Week 3: Banach Spaces and Linear
Continuous Operators
Document prepared by Anna Rozanova-Pierrat
1
1
Lecture 3.2: Banach spaces
Definition 1 A normed vector space that is complete is called a Banach space.
Example 1
1.
Rn is a Banach space for any norm de

Polycopi du cours de la 3me anne, MOA
Analyse Fonctionnelle
A. Rozanova-Pierrat
17 september 2015
Contents
Introduction
v
1 Notations
1
2 Reminders on the topology in metric and normed spaces
2.1 Distance or metric . . . . . . . . . . . . . . . . . . . .

Week 4: Hilbert spaces
Document prepared by Anna Rozanova-Pierrat1
1
Lecture 4.2: Bilinear forms
Definition 1 Let X be a vector space. We call bilinear form on X a function a : X X
such that, for all u, v, w in X and , in R , we have:
R
1. a(u + v, w) =

Week 6: Lp Spaces
Document prepared by Anna Rozanova-Pierrat1
1
Lecture 6.1: Introduction
1.1
Definitions
We have introduced Lp and Lploc -spaces in Week 2. In Week 3 we have proved that Lp for 1 p
is a Banach space (Fischer-Riesz Theorem).
Definition 1