Chapter 7 Existence Theorems 7 2 A Hilbert

Chapter 7 : Existence Theorems 7. 2 : A Hilbert Space Approach Although the discussion focuses on the Dirichlet problem , the idea are widely applicable to the study of PDE in general Vector Space S = set of objects , closed under addition and scalar multiplication Example

Chapter 7 : Existence Theorems 7. 2 : A Hilbert Space Approach Inner Product This operation associates with each pair of objects x and y In the vector space, a real number denoted <x, y>. The inner product is assumed to satisfy: Example

Chapter 7 : Existence Theorems 7. 2 : A Hilbert Space Approach Orthogonal: Example

Chapter 7 : Existence Theorems 7. 2 : A Hilbert Space Approach Norm The norm of a vector x is a real number following properties: Example with the

Chapter 7 : Existence Theorems 7. 2 : A Hilbert Space Approach Cauchy-Schwarz Inequality Example

Chapter 7 : Existence Theorems 7. 2 : A Hilbert Space Approach Length of a vector Distance between 2 vectors

Chapter 7 : Existence Theorems 7. 2 : A Hilbert Space Approach Norm induced by an inner product

Chapter 7 : Existence Theorems 7. 2 : A Hilbert Space Approach Convergent sequence Example

Chapter 7 : Existence Theorems 7. 2 : A Hilbert Space Approach Cauchy sequence Example: Remark: Every convergent sequence in V is cauchy. (proof)

Chapter 7 : Existence Theorems 7. 2 : A Hilbert Space Approach Complete Space A normed space is complete if every cauchy sequence in V is convergent. Hilbert Space A compete inner product space ( with respect the norm induced by the inner product) is called a Hilbert Space.

Chapter 7 : Existence Theorems 7. 2 : A Hilbert Space Approach Hilbert Space A compete inner product space ( with respect the norm induced by the inner product) is called a Hilbert Space. Example

Chapter 7 : Existence Theorems 7. 2 : A Hilbert Space Approach Hilbert Space A compete inner product space ( with respect the norm induced by the inner product) is called a Hilbert Space. Example Is a Hilbert space with the inner product. and norm

Chapter 7 : Existence Theorems 7. 2 : A Hilbert Space Approach Hilbert Space A compete inner product space ( with respect the norm induced by the inner product) is called a Hilbert Space. Example Is a Hilbert space with the inner product. and norm

Chapter 7 : Existence Theorems 7. 2 : A Hilbert Space Approach Hilbert Space A compete inner product space ( with respect the norm induced by the inner product) is called a Hilbert Space. Example

Chapter 7 : Existence Theorems 7. 2 : A Hilbert Space Approach The completion of V If V is an inner product space that is not complete, then V can be embedded as a dense subset of a Hilbert space H. This means that there is a Hilbert space such that V inside H. We call such H the completion of V. Example It is an inner product space with standard inner product. It is not complete ( give an example) The completion of is

Chapter 7 : Existence Theorems 7. 2 : A Hilbert Space Approach The completion of V If V is an inner product space that is not complete, then V can be embedded as a dense subset of a Hilbert space H. This means that there is a Hilbert space such that V inside H. We call such H the completion of V. Example It is an inner product space. But It is not complete The completion of is

Chapter 7 : Existence Theorems 7. 2 : A Hilbert Space Approach Linear functional A linear functional on an inner product space V is a realvalued function satisfying the linearity condition Example Bounded Linear functional A linear functional Example is bounded if there exist a positive M, such that

Chapter 7 : Existence Theorems 7. 2 : A Hilbert Space Approach Bounded Linear functional and inner product Bounded linear functionals on an inner product space V are intimately tied to the inner product on V. l ana lysis Is the other way true? ? ? Fun Riesz Representation Theorem: Every bounded linear functional on a Hilbert space can be written as an inner product with some fixed vector in the space. l in f dam enta l too If is any bounded linear functional on a Hilbert space H, then there is a unique such that unct iona Riesz Representation Theorem:

Chapter 7 : Existence Theorems 7. 2 : A Hilbert Space Approach Bilinear Form a function or functional of two variables that is linear with respect to each variable when the other variable is held fixed. Example

Chapter 7 : Existence Theorems 7. 2 : A Hilbert Space Approach Bounded Form or continuous A bilinear form a(. , . ) on a Hilbert space H is said to be continouos, if there exists a positive constant C such that Coercive Form A bilinear form a(. , . ) on a Hilbert space H is said to be coercive, if there exists a positive constant such that

Chapter 7 : Existence Theorems 7. 2 : A Hilbert Space Approach Consider the following problem Variational Problem Where H is a Hilbert space with (. , . ) inner product We want to study this problem (existence and uniquence)

Chapter 7 : Existence Theorems 7. 2 : A Hilbert Space Approach The Lax-Milgram Theorem Given: a Hilbert space H, (. , . ), a continuous, coercive bilinear form a(. , . ) and a continuous linear functional F in H’, There exists a unique u in H such that

Chapter 7 : Existence Theorems 7. 2 : A Hilbert Space Approach Example 1) Bilinear form 2) Bounded (cont. ) 3) Coercive